Equations needing more than one transformation: why the addition is undone before the multiplication, simplifying one side by combining like terms before solving, building a two-step equation from a verbal model, and checking a multi-step solution against the original equation.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 3 — Solving Linear Equations
Solving Multi-Step Equations
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.3 Solving Multi-Step Equations §3.3, pp. 144-149 — the lesson these objectives are drawn from
Warm-up
You have two solving moves. This lesson is about which to use first when both are needed.
Discussion prompt
To evaluate 3x plus 7 at x equal to 4, which operation do you perform first? Now think about solving 3x plus 7 equals 19 — which operation would you undo first?
Hint: The two answers are opposite, and that is the whole point.
Answer:
\[ \text{evaluating: } 3(4) + 7 = 12 + 7 = 19 \quad \text{multiply first} \]
\[ \text{solving: } 3x + 7 = 19 \;\rightarrow\; 3x = 12 \;\rightarrow\; x = 4 \quad \text{subtract first} \]
Evaluating multiplies then adds; solving subtracts then divides. Solving unbuilds the expression, so it works in the reverse order — the last thing done to x is the first thing undone.
Concept
When more than one operation has been applied to the variable, undo them in the reverse of the order they were applied. The addition or subtraction is on the outside, so it comes off first.
multi-step equation — An equation that requires more than one transformation to isolate the variable.
Simplify one or both sides first if anything can be combined, then use inverse operations to isolate the variable.
Figure (svg): The expression 3x plus 7 shown as two wrappings round x, undone from the outside inwards
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.3 Solving Multi-Step Equations §3.3, pp. 144-144
Section
Section 1
Concept
In an equation such as three x plus seven equals eight, two things have been done to x: it was multiplied by three and then seven was added. Undoing them in the reverse order isolates the variable in two clean steps.
Each step is licensed by a property of equality, exactly as in the two previous lessons.
Figure (svg): The equation 3x plus 7 equals 8 solved in two steps, subtracting then dividing
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.3 Solving Multi-Step Equations §3.3, pp. 144-144 — Example 1, Solve a Linear Equation
Picture it
One move per line, with the reason beside it.
Figure (svg): The equation 3x plus 7 equals 8 solved in two steps, subtracting then dividing
The layout matters as much as the moves. A solution written one move per line can be checked line by line; a solution compressed into two lines cannot.
Worked example
This is Example 1 from the textbook. Two operations, undone in reverse order.
\[ \text{Solve } \; 3x - 7 = 8. \]
Identify what has been done to x
Why: Multiplied by three, then seven subtracted. The subtraction is on the outside.
\[ \times 3\text{ then } -7 \]
Undo the subtraction first
Why: Add seven to each side, by the addition property of equality.
\[ 3 x = 15 \]
Undo the multiplication second
Why: Divide each side by three, by the division property of equality.
\[ x = 5 \]
Check in the original equation
Why: Three times five is fifteen, minus seven is eight, which matches.
\[ 8 = 8 \]
Figure (svg): A substitution check on a two-step solution, showing every stage
\[ 3x - 7 = 8 \;\Longrightarrow\; x = 5 \]
Verify: follow the check in the order of operations
Why: Substituting five gives three times five first, then minus seven — multiply before subtract, which is the order of operations from Lesson 1.3. Notice that the check runs in the opposite order to the solving, which is exactly what unwrapping means.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.3 Solving Multi-Step Equations §3.3, pp. 144-144
Ranking
Four moves, one correct sequence, for the equation 3x minus 7 equals 8.
Put in order
Why: Identifying the two operations comes first, because it determines the order of the next two. The addition or subtraction is undone before the multiplication or division, since it was applied last. The substitution check comes at the end, and it runs in the opposite order to the solving.
Worked example
Guided Practice 1 to 3. One has a negative coefficient.
\[ \text{Solve } \; 6x + 15 = 9, \quad 7x - 4 = -11, \quad -2y + 5 = 1. \]
Solve the first
Why: Subtract fifteen from both sides to get 6x equals negative six, then divide by six.
\[ x = -1 \]
Solve the second
Why: Add four to both sides to get 7x equals negative seven, then divide by seven.
\[ x = -1 \]
Solve the third
Why: Subtract five from both sides to get negative 2y equals negative four, then divide by negative two.
\[ y = 2 \]
Note the pattern of moves
Why: In every case the constant came off first and the coefficient second.
Figure (svg): The solution to Worked example three from guided practice shown as a ladder of expressions, one row per algebraic move
\[ x = -1, \quad x = -1, \quad y = 2 \]
Verify: substitute each answer back
Why: Six times negative one plus fifteen is nine; seven times negative one minus four is negative eleven; negative two times two plus five is one. All three originals come out true, and the third confirms that dividing by a negative coefficient handles its own sign.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.3 Solving Multi-Step Equations §3.3, pp. 144-144
Trap
\[ 3x + 7 = 22 \]
Divide both sides by 3 first, since the 3 is attached to the x
Why: The coefficient is the thing physically touching the variable, so it looks like the closest obstacle.
\[ x + \tfrac{7}{3} = \tfrac{22}{3} \]
Legal, and the numbers have become fractions with two steps still to go. The equation is no simpler than it was.
\[ 3x + 7 - 7 = 22 - 7 \;\Longrightarrow\; 3x = 15 \;\Longrightarrow\; x = 5 \]
Take off the outermost operation first, which is the addition
Why: The seven was added after the multiplication, so it is the outer wrapping and comes off first.
Both routes give x equal to five, so this is about effort rather than legality. Subtracting first keeps the numbers whole, which is why it is the standard order.
Sorting
For each equation, decide which operation to undo first.
Sort into buckets
Sort each equation by the first move.
The two-step equations all reduce to one-step equations after the first move. That is the whole strategy: peel one operation off and you are back in familiar territory.
Faded example
The first move is done. Complete the second.
Fill in the blanks
3x - 7 = 8 \;\rightarrow\; 3x = 15 \;\rightarrow\; x = 5
Why: Adding seven to both sides gives three x equals fifteen, and dividing both sides by three gives x equal to five. Each blank is one complete transformation, and writing them as separate lines is what makes a wrong step findable later.
Elimination
The equation is 4x plus 6 equals 26.
Eliminate the wrong options
Which first move keeps the arithmetic simplest?
Survives elimination: A
Why: Subtracting six gives four x equals twenty, a clean one-step equation with whole numbers. All four options are legal transformations, and only one of them both makes progress and keeps the arithmetic simple — which is the standard by which a move is chosen once legality is established.
Section
Section 2
Concept
The order of operations from Lesson 1.3 says how an expression is built: powers, then multiplication and division, then addition and subtraction. Solving takes the expression apart, so it works up that list from the bottom.
That is the same reasoning as taking off a coat before a jumper: the outermost layer comes off first.
Figure (svg): The unwrapping order shown as the order of operations reversed
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.3 Solving Multi-Step Equations §3.3, pp. 144-144 — the instruction to use inverse operations to isolate the variable
Picture it
The right column is the left column read from the bottom up.
Figure (svg): The unwrapping order shown as the order of operations reversed
Knowing this relationship means you never have to memorise the solving order separately. It is the order of operations, reversed.
Worked example
Dividing first is legal. Comparing the two routes shows why nobody does it.
\[ \text{Solve } \; 3x + 7 = 22 \; \text{ twice, once dividing first and once subtracting first.} \]
Route one: divide both sides by 3 first
Why: Every term on both sides must be divided, giving x plus seven thirds equals twenty-two thirds.
\[ x + \frac{7}{3} = \frac{22}{3} \]
Finish route one
Why: Subtract seven thirds from both sides: twenty-two thirds minus seven thirds is fifteen thirds, which is five.
\[ x = 5 \]
Route two: subtract 7 from both sides first
Why: Three x equals fifteen, with whole numbers throughout.
\[ 3 x = 15 \]
Finish route two
Why: Divide both sides by three.
\[ x = 5 \]
Figure (svg): Two columns contrasting undoing the addition first with undoing the multiplication first
\[ 3x + 7 = 22 \;\Longrightarrow\; x = 5 \]
Verify: compare the arithmetic of the two routes
Why: Both reach five, so both are correct. The first route required three fraction calculations and the second required none, which is the entire argument for the standard order — it is about effort rather than validity.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Stage | Building the expression | Solving the equation |
|---|---|---|
| first | powers | undo addition and subtraction |
| second | multiply and divide | undo multiplication and division |
| third | add and subtract | undo powers |
Reading the two columns shows them as mirror images. Nothing about the solving order has to be memorised separately once that relationship is seen.
Worked example
With three layers the ordering matters more, and the same rule handles it.
\[ \text{Solve } \; \tfrac{x}{4} + 2 = 6. \]
Identify the operations on x
Why: x is divided by four, and then two is added. The addition is outermost.
\[ \div 4\text{ then } +2 \]
Undo the addition first
Why: Subtract two from both sides.
\[ \frac{x}{4} = 4 \]
Undo the division second
Why: Multiply both sides by four.
\[ x = 16 \]
Check in the original
Why: Sixteen over four is four, plus two is six.
\[ 6 = 6 \]
Figure (svg): The solution to Worked example a three-operation equation shown as a ladder of expressions, one row per algebraic move
\[ \tfrac{x}{4} + 2 = 6 \;\Longrightarrow\; x = 16 \]
Verify: check that the order was right by trying the alternative
Why: Multiplying by four first would give x plus eight equals twenty-four, since the two must be multiplied as well — and that still solves to sixteen. Legal again, and it required multiplying every term rather than just one, which is the extra work the standard order avoids.
Error analysis
The student solved two equations, both by dividing first. One answer is wrong.
Annotate
On: \( 3x + 7 = 22 \;\rightarrow\; x + 7 = \tfrac{22}{3} \qquad \tfrac{x}{4} + 2 = 6 \;\rightarrow\; x + 8 = 24 \)
Both errors and both correct routes are caught by the same check: substitute the answer into the original equation. Only one of these two students' answers survives it.
Prediction
Both routes are legal. One of them is much less work.
Predict first
In solving 5x plus 20 equals 45, which first move keeps every number whole?
Correct: Both keep the numbers whole here.
\[ 5x + 20 = 45 \;\rightarrow\; 5x = 25 \;\rightarrow\; x = 5 \]
\[ 5x + 20 = 45 \;\rightarrow\; x + 4 = 9 \;\rightarrow\; x = 5 \]
Why: Subtracting twenty gives 5x equals 25, and dividing by five gives x plus four equals nine — both perfectly clean, because twenty and forty-five are both divisible by five. That is a coincidence of these particular numbers, and it is worth noticing: the standard order is preferred because it works cleanly always, not because the alternative always fails.
Elimination
A student divides both sides of 3x plus 7 equals 22 by three and writes x plus 7 equals 22 over 3.
Eliminate the wrong options
What is the error?
Survives elimination: A
Why: Dividing a side by three means dividing every term on it, so the seven should have become seven thirds. Leaving one term undivided breaks the balance just as surely as ignoring a whole side would, and it is the specific fragility that makes dividing first riskier than subtracting first.
Socratic
The unwrapping picture depends on knowing which layer is outside.
Discussion prompt
In the expression 3x plus 7, explain how you can tell that the multiplication happened before the addition, using the order of operations. Then say how the answer would change for the expression 3 times the quantity x plus 7.
Hint: Evaluate both at a specific value and watch the order.
Answer:
Evaluating 3x plus 7 at x equal to 4 means multiplying first, by the order of operations, and adding second — twelve then nineteen. So the multiplication is the inner layer and the addition the outer one, and solving reverses that.
\[ 3(x + 7) \text{ at } x = 4: \quad 3(11) = 33 \]
With brackets the addition happens first and the multiplication second, so the layers swap. Solving three times the quantity x plus seven equals thirty-three would divide by three first and subtract seven second — the reverse order again, but of a different building sequence. Reading the brackets is what tells you which case you are in.
Section
Section 3
Concept
If a side of the equation has like terms, combine them before applying any inverse operation. Simplifying one side is not a transformation of the equation — it changes how a side is written without changing what it equals.
Because simplifying changes only how a side is written, it does not need to be balanced by anything on the other side.
Figure (svg): An equation with like terms on one side, combined before any inverse operation is applied
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.3 Solving Multi-Step Equations §3.3, pp. 145-145 — Example 3, Combine Like Terms First
Picture it
The first move is on one side only, and the rest are on both.
Figure (svg): An equation with like terms on one side, combined before any inverse operation is applied
Notice that the first line's move touches only the left side, while the next two touch both. That difference is exactly what distinguishes simplifying from transforming.
Worked example
This is Example 3 from the textbook.
\[ \text{Solve } \; 7x - 3x - 8 = 24. \]
Combine the like terms on the left
Why: Seven x and negative three x combine to four x. Nothing on the right side changes.
\[ 4 x - 8 = 24 \]
Undo the subtraction
Why: Add eight to each side.
\[ 4 x = 32 \]
Undo the multiplication
Why: Divide each side by four.
\[ x = 8 \]
Check in the original equation
Why: Seven times eight minus three times eight minus eight is 56 minus 24 minus 8, which is 24.
\[ 24 = 24 \]
Figure (svg): An equation with like terms on one side, combined before any inverse operation is applied
\[ 7x - 3x - 8 = 24 \;\Longrightarrow\; x = 8 \]
Verify: check against the uncombined original
Why: Substituting eight into the original, before any combining, gives 56 minus 24 minus 8, which is 24. Checking against the equation as given rather than the simplified version is what would catch an error in the combining step itself.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.3 Solving Multi-Step Equations §3.3, pp. 145-145
Sorting
One kind of move touches one side; the other must touch both.
Sort into buckets
Sort each move by whether it must be applied to both sides.
The test is always the same: does the move change what the side is worth? If not, it needs no partner on the other side.
Worked example
Brackets are removed before like terms are combined, exactly as in Lesson 2.7.
\[ \text{Solve } \; 2(x + 3) + 4x = 24. \]
Distribute over the bracket
Why: Two x plus six, added to four x.
\[ 2 x + 6 + 4 x = 24 \]
Combine the like terms
Why: Two x and four x make six x, and the six is left alone.
\[ 6 x + 6 = 24 \]
Undo the addition
Why: Subtract six from both sides.
\[ 6 x = 18 \]
Undo the multiplication
Why: Divide both sides by six.
\[ x = 3 \]
Figure (svg): The solution to Worked example a bracket to distribute first shown as a ladder of expressions, one row per algebraic move
\[ 2(x + 3) + 4x = 24 \;\Longrightarrow\; x = 3 \]
Verify: substitute 3 into the original with the bracket intact
Why: Two times six plus twelve is twelve plus twelve, which is 24. Checking against the bracketed original rather than the distributed version is what tests the distribution as well as the solving.
Trap
\[ 7x - 3x - 8 = 24 \]
Combine 7x and -3x on the left, then subtract 3x from the right as well to keep the balance
Why: Every other move in this chapter had to be applied to both sides, so this one looks as though it should be too.
\[ 4x - 8 = 24 - 3x \quad \text{(wrong)} \]
Nothing was taken away from the left side. Seven x minus three x is another way of writing four x, and rewriting a side does not change its value.
\[ 7x - 3x - 8 = 24 \;\Longrightarrow\; 4x - 8 = 24 \]
Simplify a side without touching the other, because simplifying changes only the writing
Why: A transformation changes the value of both sides equally; a simplification changes the value of neither.
The test is whether the side's value changed. Combining like terms leaves it identical for every value of x, so there is nothing to balance.
Ranking
Five moves, one correct sequence, for an equation with brackets and like terms.
Put in order
Why: Brackets come off first, because a term inside one is not yet a term of the side. Like terms are combined second, which needs the brackets gone to be visible. Then the two transforming moves in the usual order, and the check last. The first two moves simplify sides; the next two transform the equation.
Faded example
The like terms are combined. Complete the solving.
Fill in the blanks
7x - 3x - 8 = 24 \;\rightarrow\; 4x - 8 = 24 \;\rightarrow\; 4x = 32 \;\rightarrow\; x = 8
Why: Combining seven x and negative three x gives four x, which stays four x through the next line since adding eight to both sides does not touch the coefficient. Dividing thirty-two by four gives eight. The repeated blank is deliberate: the coefficient survives every step until the division that removes it.
Socratic
It is the one move in the chapter that touches a single side.
Discussion prompt
Explain why combining like terms on one side does not require any change to the other side, and contrast it with subtracting a number from one side. Use the idea of what each move does to the side's value.
Hint: Substitute a value and compare the side before and after each move.
Answer:
Combining like terms rewrites a side into an equal expression: seven x minus three x and four x give the same number for every value of x, so the side is worth exactly what it was worth before. Nothing changed, so nothing needs compensating.
Subtracting a number is different: the side is genuinely worth less afterwards. To keep the two sides equal, the other side must lose the same amount. The distinction is between changing how a side is written and changing what it is worth, and only the second kind of move needs a partner.
Section
Section 4
Concept
Almost every two-step equation from a real situation has the same shape: something fixed, plus something per unit, equalling a total. The fixed part comes off first and the rate is divided out second.
\[ \text{total} = \text{start} + \text{rate} \times \text{quantity} \]
This is the shape from Lesson 1.1 and Lesson 1.6, now with the unknown inside the equation rather than at the end of a calculation.
Figure (svg): Earth's crust with temperature rising 30 degrees per kilometre from a surface temperature of 24 degrees
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.3 Solving Multi-Step Equations §3.3, pp. 145-145 — Example 2, Use a Verbal Model
Picture it
Twenty-four degrees at the surface, rising thirty degrees per kilometre.
Figure (svg): Earth's crust with temperature rising 30 degrees per kilometre from a surface temperature of 24 degrees
The surface temperature is the fixed part and the thirty degrees per kilometre is the rate. Finding the depth means subtracting the fixed part and then dividing by the rate.
Worked example
Example 2 from the textbook, worked through the full plan from Lesson 1.6.
\[ \text{Solve } \; 114 = 24 + 30d \; \text{ for the depth } d \text{ in kilometres.} \]
Write the verbal model
Why: The temperature inside Earth equals the surface temperature plus the rate of increase times the depth.
Assign labels
Why: 114 degrees, 24 degrees, 30 degrees per kilometre, and d kilometres.
Write and solve the algebraic model
Why: Subtract 24 from each side to get 90 equals 30d.
\[ 90 = 30 d \]
Divide and answer in words
Why: Divide each side by 30 to get d equals 3, so the temperature reaches 114 degrees at a depth of 3 kilometres.
\[ 3\text{ kilometres} \]
Figure (svg): Earth's crust with temperature rising 30 degrees per kilometre from a surface temperature of 24 degrees
\[ 114 = 24 + 30d \;\Longrightarrow\; d = 3 \text{ kilometres} \]
Verify: rebuild the temperature from the answer
Why: At three kilometres the increase is thirty times three, which is ninety degrees, and adding the surface temperature of twenty-four gives 114 — the figure the problem asked about. Recovering a given number is the strongest form of check.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.3 Solving Multi-Step Equations §3.3, pp. 145-145
Translation
Four situations. Let x be the unknown in each.
Match the pairs
Why: All four have the same shape: a fixed starting value, a rate multiplying the unknown, and a total. Recognising that shape is worth more than any individual translation, because the solving is then automatic — subtract the fixed part, divide by the rate.
Worked example
Guided Practice 4. The model is reused with one number changed.
\[ \text{At what depth is the temperature } 174 \text{ degrees?} \]
Reuse the verbal model and labels
Why: Only the temperature inside Earth changes; the surface temperature and the rate are unchanged.
\[ 174 = 24 + 30 d \]
Subtract the surface temperature
Why: 174 minus 24 is 150.
\[ 150 = 30 d \]
Divide by the rate
Why: 150 over 30 is 5.
\[ d = 5 \]
Answer in words with the unit
Why: Five kilometres below the surface.
\[ 5\text{ kilometres} \]
Figure (svg): The solution to Worked example the same model at a different temperature shown as a ladder of expressions, one row per algebraic move
\[ 174 = 24 + 30d \;\Longrightarrow\; d = 5 \text{ kilometres} \]
Verify: compare with the previous answer
Why: The temperature rose by sixty degrees from 114 to 174, and sixty degrees at thirty degrees per kilometre is two more kilometres — which takes three to five, exactly as computed. Comparing two answers from the same model is a check the model itself makes possible.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.3 Solving Multi-Step Equations §3.3, pp. 145-145
Trap
\[ 114 = 24 + 30d \]
Divide 114 by 30 to find the depth
Why: Both numbers are in the problem and dividing a total by a rate usually gives a quantity.
\[ d = 3.8 \quad \text{(wrong)} \]
The surface temperature of twenty-four degrees was there before any depth was reached, so it is not part of what the rate produced. Dividing it by the rate credits it to the descent.
\[ 114 - 24 = 30d \;\Longrightarrow\; 90 = 30d \;\Longrightarrow\; d = 3 \]
Remove the fixed part before dividing by the rate
Why: The rate only accounts for the increase, so the starting value has to come off first.
This is exactly why the addition is undone before the multiplication. The order of the moves and the meaning of the model agree with each other.
Elimination
A taxi charges 3 dollars to get in plus 2 dollars a mile. A journey cost 19 dollars.
Eliminate the wrong options
Which equation gives the number of miles?
Survives elimination: A
Why: The three dollars is paid once so it is added, and the two dollars is paid per mile so it multiplies m. Solving gives eight miles. Testing at zero miles settles it: a journey of no miles should cost the three dollar boarding charge, and only the first option gives that.
Missing information
A question can be perfectly well written and still be unanswerable.
Discussion prompt
The temperature inside Earth's crust rises 30 degrees per kilometre. At what depth is it 114 degrees? Say exactly what is missing, and give the two different answers you would get with a surface temperature of 24 degrees and of 0 degrees.
Hint: Write the verbal model and see which quantity has no number.
Answer:
The surface temperature is missing. The model needs a starting value as well as a rate, and without it the fixed part of the equation is unknown.
\[ \text{surface } 24: \; 114 = 24 + 30d \Rightarrow d = 3 \qquad \text{surface } 0: \; 114 = 30d \Rightarrow d = 3.8 \]
The two answers differ by nearly a kilometre. A missing starting value is one of the easiest things to overlook in a rate problem and one of the most damaging, since it distorts every answer rather than just one — the same warning as the phone-call problem in Lesson 1.5.
Socratic
The algebraic order and the physical meaning agree, and that is not a coincidence.
Discussion prompt
Explain why subtracting the surface temperature before dividing by the rate makes sense physically as well as algebraically. Then describe what the intermediate value of 90 represents in the situation.
Hint: Ask what the ninety is measuring.
Answer:
The rate of thirty degrees per kilometre accounts only for the increase caused by going deeper. The twenty-four degrees was already there at zero depth, so it has to be removed before the remainder can be attributed to the descent.
The ninety is the increase in temperature caused by the depth alone, measured in degrees. Dividing it by thirty degrees per kilometre gives kilometres, and the units confirm the reasoning: degrees divided by degrees per kilometre leaves kilometres. Every intermediate line in a well-set-up solution means something, and being able to say what is a good sign the model was built correctly.
Section
Section 5
Concept
A multi-step solution is checked by substituting into the original equation and evaluating it by the order of operations — which is the reverse of the order the solving used.
Because the check reverses the solving, an error in the solving is very unlikely to survive it.
Figure (svg): A substitution check on a two-step solution, showing every stage
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.3 Solving Multi-Step Equations §3.3, pp. 144-144 — the CHECK step of Example 1
Picture it
Multiply first, then subtract — the opposite of how the answer was found.
Figure (svg): A substitution check on a two-step solution, showing every stage
The verdict is true or false, with nothing in between. A check that comes out nearly right is a check that failed.
Worked example
The check from Example 1, written out in full.
\[ \text{Check that } x = 5 \text{ solves } \; 3x - 7 = 8. \]
Write the original equation
Why: Not any of the lines produced while solving.
\[ 3 x - 7 = 8 \]
Substitute 5 for x
Why: The five goes wherever x appeared.
\[ 3(5) - 7 = 8 \]
Evaluate by the order of operations
Why: Multiply before subtracting: three fives are fifteen.
\[ 15 - 7 = 8 \]
Read off the verdict
Why: Fifteen minus seven is eight, which matches the right side.
Figure (svg): A substitution check on a two-step solution, showing every stage
\[ 3(5) - 7 = 15 - 7 = 8 \;\checkmark \]
Verify: notice the order the check used
Why: The check multiplied first and subtracted second, while the solving added first and divided second. The two run in opposite directions, which is what makes the check an independent test rather than a repetition of the solving.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.3 Solving Multi-Step Equations §3.3, pp. 144-144
Elimination
A student solved 2(x plus 3) plus 4x equals 24 and got x equals 3.
Eliminate the wrong options
Which substitution tests every step?
Survives elimination: A
Why: Only the original equation is untouched by the student's work, so only it can catch an error in the distribution or the combining. Substituting three gives two times six plus twelve, which is twenty-four — confirming every step at once.
Worked example
A student solves in the wrong order and gets a plausible answer.
\[ \text{A student solves } 4x + 6 = 26 \text{ and gets } x = 8. \text{ Check it.} \]
Substitute 8 into the original
Why: Four times eight is thirty-two, plus six is thirty-eight.
Judge
Why: Thirty-eight is not twenty-six, so eight is not a solution.
Diagnose
Why: The student appears to have divided 26 by 4 and then added, or subtracted after dividing, rather than subtracting first.
Solve correctly
Why: Subtract six to get 4x equals 20, then divide by four.
\[ x = 5 \]
Figure (svg): The solution to Worked example a check that catches an error shown as a ladder of expressions, one row per algebraic move
\[ 4(8) + 6 = 38 \neq 26 \qquad 4(5) + 6 = 26 \;\checkmark \]
Verify: check the corrected answer
Why: Four times five is twenty, plus six is twenty-six, which matches. The corrected solution passes the check the wrong one failed, which confirms the diagnosis rather than merely replacing one guess with another.
Trap
\[ 7x - 3x - 8 = 24 \;\rightarrow\; 4x - 8 = 24 \;\rightarrow\; x = 8 \]
Check by substituting 8 into 4x minus 8 equals 24
Why: That line is simpler than the original and the substitution is easier.
If the combining step was wrong, the middle line inherits the error and the check passes anyway. It tests only the last two steps.
\[ 7(8) - 3(8) - 8 = 56 - 24 - 8 = 24 \;\checkmark \]
Substitute into the equation as it was given, before any simplification
Why: The original is the only line you did not write, so it is the only one that tests everything.
A check is worth doing only if it could fail. Checking against your own working can fail only in the steps after that line.
Sorting
Substitute each claimed solution into its own original equation.
Sort into buckets
Sort each claim by whether the solution is correct.
Every failure here was exposed by a single substitution taking a few seconds. Three of six claimed solutions were wrong, which is a realistic rate for unchecked multi-step work.
Faded example
The substitution is made. Evaluate by the order of operations.
Fill in the blanks
\text15 3x - 7 = 8 \text8 x = 5: \; 3(5) - 7 = ___ - 7 = ___ \;\checkmark
Why: The order of operations multiplies before subtracting, so three times five is fifteen and then fifteen minus seven is eight, matching the right side. Notice that this order is the reverse of the solving order, which added before dividing — that reversal is what makes the check independent.
Socratic
The check and the solving use the same equation and different reasoning.
Discussion prompt
Explain why substituting into the original equation is a genuinely independent test of a multi-step solution, referring to the order in which each process works. Then say what kind of error it would still miss.
Hint: Compare which operation each process performs first.
Answer:
The solving works from the outside in, undoing the addition and then the multiplication. The check works from the inside out, performing the multiplication and then the subtraction, because it follows the order of operations. Different operations in a different order means an error in one is very unlikely to be repeated identically in the other.
What it would still miss is an error in copying the original equation itself. If the equation was written down wrongly at the start, both the solving and the check use the wrong equation and both agree. That is why the first thing to re-read when an answer seems strange is the question rather than the working.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| One-step | Two-step | |
|---|---|---|
| Operations on the variable | one | two |
| First move | undo the only operation | undo the addition or subtraction |
| Second move | none — check the answer | undo the multiplication or division |
The second move of a two-step equation is exactly a one-step equation. Every multi-step problem reduces to a familiar one after the first move.
Pattern
Whether the equation has brackets, like terms, or both, the same five moves cover it.
Steps one and two touch one side at a time; steps three and four must touch both. Knowing which kind of move you are making is what stops a simplification being wrongly balanced.
OpenStax Elementary Algebra 2e, §2.4 Use a General Strategy to Solve Linear Equations §2.4
Check
A two-step equation. Undo the constant first.
Check your understanding
Solve 5x plus 8 equals 33.
Answer: A
Why: Subtracting eight from both sides gives 5x equals 25, and dividing by five gives x equal to five. Substituting confirms it: five fives are twenty-five, plus eight is thirty-three.
Check
Like terms first. Simplify one side before transforming.
Check your understanding
Solve 9x minus 4x plus 3 equals 23.
Answer: A
Why: Combining nine x and negative four x gives five x, so the equation is five x plus three equals twenty-three. Subtracting three gives five x equals twenty, and dividing by five gives four. Substituting confirms it: thirty-six minus sixteen plus three is twenty-three.
Check
A word problem. Identify the fixed part and the rate.
Check your understanding
A gym charges a 40 dollar joining fee plus 25 dollars a month. After how many months has a member paid 190 dollars?
Answer: A
Why: The equation is 40 plus 25m equals 190. Subtracting forty gives 25m equals 150, and dividing by twenty-five gives six months. Checking rebuilds the total: six months at twenty-five is 150, plus the forty dollar fee is 190.
Real world
A phone plan costs 18 dollars a month plus 4 cents a minute. One month's bill was 31.20 dollars.
Discussion prompt
Write and solve an equation for the number of minutes used, being careful with units. Then say which of your two solving steps corresponds to removing the fixed charge, and what the intermediate value represents in the situation.
Hint: Convert the four cents to dollars before combining it with the monthly fee.
Answer:
\[ 18 + 0.04m = 31.20 \]
\[ 0.04m = 13.20 \;\Longrightarrow\; m = 330 \text{ minutes} \]
The first step, subtracting eighteen, removes the fixed monthly charge and leaves 13.20 — the amount spent on calls alone. The second step divides that by the per-minute rate to convert dollars into minutes, and the units confirm it: dollars divided by dollars per minute leaves minutes.
Converting the four cents to 0.04 dollars before writing the equation is essential, exactly as in Lesson 1.5. Adding cents to dollars would have made the equation wrong before any solving began.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
In solving 3x plus 7 equals 22, is it wrong to divide by 3 first?
Correct: No, it is legal but produces fractions and more work.
\[ 3x + 7 = 22 \;\rightarrow\; x + \tfrac{7}{3} = \tfrac{22}{3} \;\rightarrow\; x = \tfrac{15}{3} = 5 \]
\[ 3x + 7 = 22 \;\rightarrow\; 3x = 15 \;\rightarrow\; x = 5 \]
Why: Dividing both sides by three is a valid transformation and reaches x equal to five, provided every term is divided — the seven becomes seven thirds and the twenty-two becomes twenty-two thirds. The standard order is preferred because it keeps whole numbers and because forgetting to divide one of the terms is an easy mistake to make. Legality and convenience are different questions, and it is worth knowing which one the rule is about.
Explain it
They can solve one-step equations and freeze when two operations appear.
Discussion prompt
In no more than four sentences, explain which operation to undo first when two have been applied, using an everyday image rather than a rule. Then tell them the one check that proves they got the order right.
Hint: Think about getting dressed and undressed.
Answer:
A usable answer: think of the operations as layers of clothing. The multiplication went on first and the addition went on over the top, so the addition comes off first. You always take off the outermost layer, which means undoing whatever was done last.
The check is to substitute the answer into the equation you were given and work it out using the ordinary order of operations. If it comes out true, the order was right; and notice that the check multiplies before adding, which is the opposite of the order you solved in — that reversal is the whole idea.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: The order is fixed by asking which operation was applied last, since that is the outer layer. Combining first is fixed by scanning each side for like terms before touching anything. The one-side-or-both question is fixed by asking whether the move changes what the side is worth. Word problems are fixed by looking for the fixed part and the rate, which almost every two-step situation has. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
At the top of a page draw the expression 3x plus 7 as two nested boxes round an x, labelling the inner box multiply by three and the outer box add seven, with arrows showing which is undone first. Underneath, solve one two-step equation writing one move per line with its reason beside it, and show the substitution check against the original. In the middle, write two columns headed simplifying and transforming, and put three moves under each with a note on why only one column needs both sides. Near the bottom, write one word problem with a fixed part and a rate, its equation, and its solution in words with a unit. Finally, in the margin, write the building order and the solving order side by side and draw an arrow showing that one is the other reversed.
Your two columns should differ on exactly one test: does the move change what the side is worth? If a move you listed under simplifying does change the side's value, it belongs in the other column.
Recap
Five things, and the first one is the decision the whole lesson turns on.
| If the question says | Your first move is |
|---|---|
| 3x - 7 = 8 | Add 7 to both sides |
| 7x - 3x - 8 = 24 | Combine the like terms first |
| 2(x + 3) + 4x = 24 | Distribute, then combine |
| 24 + 30d = 114 | Subtract the fixed part, then divide by the rate |
| Check your solution | Substitute into the original, brackets and all |
Lesson 3.4 adds one more complication: the variable appearing on both sides of the equation. The strategy is to gather the variable terms on one side first, and then the equation becomes one of the kind you have just solved.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 3 Solving Linear Equations — Lesson 3.3 Solving Multi-Step Equations §3.3, pp. 144-149 — everything on these slides traces back here
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