Chapter 3 of Algebra 1: Concepts and Skills, built for a visual learner. Inverse operations on a balance scale, unwrapping multi-step equations in reverse, variables on both sides plus the no-solution and infinite-solution cases, clearing fractions and decimals, rearranging formulas, and ratio, rate and percent problems drawn as bar models.
Subject: Algebra 1 · 61 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
Algebra 1 · Chapter 3
Undoing operations, keeping the balance, and turning formulas, ratios and percents into equations you can solve
Objectives
Chapter 2 taught you to simplify. This chapter teaches you to solve — to find the number the letter was hiding.
Figure (svg): A chain showing x having 7 added to it to become x plus 7, and the reverse arrow subtracting 7 to get back to x
Section
Section 3.1
Concept
An equation stays true as long as both sides change identically. That single permission is the whole of solving.
Figure (svg): Three balance scales in a row: x plus 7 equals 12, then 7 removed from both pans, then x equals 5, staying level throughout
inverse operations — Two operations that undo each other: adding and subtracting, multiplying and dividing. Solving means applying the inverse of whatever was done to the variable.
Prediction
Commit before working.
\[ x + 7 = 12 \]
Predict first
What is the single move that leaves x alone on the left?
Correct: Subtract 7 from both sides.
Why: Seven is being added to x, so the inverse is to subtract seven. Doing it to both sides keeps the scale level, giving x equals five. Adding seven would make it worse, producing x plus fourteen, and dividing does not undo an addition at all.
Worked example
Solve the equation below.
\[ x + 7 = 12 \]
Subtract 7 from both sides
Why: Seven is added to x, so subtracting seven is the inverse. Doing it to both sides is what keeps the equation true rather than merely convenient.
\[ x + 7 - 7 = 12 - 7 \]
Simplify each side
Why: On the left the plus seven and minus seven cancel, leaving x alone, which is the goal.
\[ x = 5 \]
Figure (svg): Three balance scales in a row: x plus 7 equals 12, then 7 removed from both pans, then x equals 5, staying level throughout
Verify: substitute 5 back into the original equation
Why: Five plus seven is twelve, which matches the right side exactly, so five really is the solution rather than an arithmetic accident.
Error analysis
Find what went wrong here.
Annotate
On: \( x + 7 = 12 \;\overset{?}{\Longrightarrow}\; x = 12 + 7 = 19 \)
There is a legitimate shortcut here — moving a term across and flipping its sign — but it is a summary of doing the same thing to both sides, not a replacement for it.
Explain it
A younger student asks why you are not allowed to just move a number across the equals sign.
Discussion prompt
Explain, in two sentences and without the word inverse, why both sides must change together.
Hint: What does the equals sign actually claim?
Answer:
Something like: the equals sign is a promise that the two sides are the same amount, so if you change one side and not the other, the promise breaks and the equation is no longer about the same problem.
Moving a term across and flipping its sign is allowed, but only because it is a shorthand for doing the same subtraction to both sides. It is a summary of the rule, not an exception to it.
Section
Section 3.2
Concept
When a number is multiplying the variable, divide both sides by it.
\[ 4x = 20 \quad \Longrightarrow \quad \frac{4x}{4} = \frac{20}{4} \quad \Longrightarrow \quad x = 5 \]
Figure (svg): A balance scale with four x chips on one side and twenty on the other, then split into four equal groups to show one x is five
Worked example
Solve the equation below, where the variable is being divided rather than multiplied.
\[ \frac{x}{3} = 8 \]
Identify what is happening to x
Why: The x is being divided by three, so the inverse move is to multiply by three.
Multiply both sides by 3
Why: Multiplying the left by three cancels the division; multiplying the right by three keeps the equation balanced.
\[ 3 \cdot \frac{x}{3} = 3 \cdot 8 \]
\[ x = 24 \]
Figure (svg): A bar split into three equal parts of eight each, showing the whole is twenty four
Verify: substitute 24 into the original
Why: Twenty-four divided by three is eight, which matches the right side, so the multiplication was the correct inverse.
Sorting
Do not solve. Just name the single move that isolates the variable.
Sort into buckets
Sort each equation by the move it needs.
Two truths and a lie
Three claims about dividing both sides. Knock out the false one.
Eliminate the wrong options
Which statement about solving by division is false?
Survives elimination: C
Why: Keep the false statement, which is C. Dividing both sides means dividing everything on each side, not just the variable term. Dividing 4x plus 8 equals 20 by four must give x plus two equals five, and skipping the eight would give a completely different equation.
Edge cases
The same move, but the sign of the answer is where people slip.
Parameter explorer
Slide the coefficient. Where does the solution of the equation a times x equals 12 go as a passes through zero?
\[ {a}x = 12 \]
Section
Section 3.3
Concept
When two things were done to x, undo the last one first. This is the order-of-operations ladder run backwards.
Figure (svg): A wrapping diagram showing x being multiplied by three then having four added, with the unwrapping arrows going the opposite way
Forwards you multiply then add. Backwards you subtract then divide.
Ranking
For the equation below, rank the moves in the order they should happen.
\[ 5x - 8 = 27 \]
Put in order
Why: The last thing done to x going forwards was subtracting eight, so that is undone first, giving 5x equals 35. Only then is the coefficient stripped by dividing, giving x equals seven. Dividing first would force you to divide the eight and the twenty-seven as well, which is legal but far messier.
Worked example
Solve the equation below.
\[ 3x + 4 = 19 \]
Undo the addition first
Why: Addition was the last operation applied to x going forwards, so it is the first one undone. Subtract four from both sides.
\[ 3x = 15 \]
Undo the multiplication second
Why: Now only the coefficient stands between you and x. Divide both sides by three.
\[ x = 5 \]
Step through it
Watch the equation shrink one step at a time. What stays true in every frame?
Verify: substitute 5 into the original equation
Why: Three times five is fifteen, and fifteen plus four is nineteen, which matches the right side, so five is genuinely the solution.
Trap
Solve three x plus four equals nineteen, starting with the coefficient.
Divide everything by 3, but only where the 3 is visible
Why: The three is attached to the x, so it is tempting to divide only that term and leave the four alone.
\[ 3x + 4 = 19 \;\to\; x + 4 = \frac{19}{3} \]
The left side lost its three but the four never did, so the two sides were not changed identically and the equation is now false.
Solve the same equation by undoing the last operation first.
Subtract 4 from both sides, then divide both sides by 3
Why: Clearing the constant first means that when you do divide, there is only a single term on the left to divide.
\[ 3x = 15 \;\Longrightarrow\; x = 5 \]
Dividing first is still legal, but only if every term on both sides is divided — and that is exactly the step people forget.
Fill the middle
Complete the middle step.
Fill in the blanks
2(x + 5) = 24 \;\Longrightarrow\; 2x + 10 = 24 \;\Longrightarrow\; 2x = 14 \;\Longrightarrow\; x = 7
Why: Distributing turns the bracket into 2x plus 10. Subtracting ten from both sides gives 2x equals 14, and dividing by two gives seven. There is a faster route here — divide both sides by two first, giving x plus five equals twelve — and it works because the whole left side is a single product.
Error analysis
This solution reaches a wrong answer. Find the exact line where it breaks.
Annotate
On: \( \begin{aligned} 6x - 5 &= 19 \\ 6x &= 14 \\ x &= \tfrac{7}{3} \end{aligned} \)
The habit that catches this: say the operation out loud before writing it. There is a minus five, so add five.
Faded example
Same skill, fewer given lines.
Fill in the blanks
\frac945 - 2 = 7 \;\Longrightarrow\; \frac______ = ___ \;\Longrightarrow\; x = ___
Why: Adding two to both sides clears the constant, leaving x over five equals nine. Multiplying both sides by five then rebuilds the whole, giving forty-five. Dividing by five at the end instead of multiplying is the classic slip and would give nine fifths.
Section
Section 3.4
Concept
When x appears on both sides, remove it from one side by subtracting it from both. Then it is an ordinary two-step equation.
Figure (svg): A balance scale with x chips on both pans, showing two x removed from each side to gather the variable on one side
Choosing the side with the smaller coefficient keeps the numbers positive, which is a real convenience rather than a rule.
Worked example
Solve the equation below.
\[ 5x + 2 = 2x + 11 \]
Subtract 2x from both sides
Why: Two x is the smaller variable amount, so removing it clears the right side and leaves the left coefficient positive.
\[ 3x + 2 = 11 \]
Subtract 2 from both sides
Why: Now it is an ordinary two-step equation, so undo the addition next.
\[ 3x = 9 \]
Divide both sides by 3
Why: The last wrapper is the coefficient.
\[ x = 3 \]
Figure (svg): A balance scale with x chips on both pans, showing two x removed from each side to gather the variable on one side
Verify: substitute 3 into both sides separately
Why: The left gives fifteen plus two, which is seventeen. The right gives six plus eleven, which is also seventeen. Both sides agree, so three is the solution.
Anomaly
Try to solve it and watch what happens to x.
\[ 2x + 5 = 2x + 9 \]
Predict first
What happens when you subtract 2x from both sides, and what does that mean?
Correct: You get 5 equals 9, which is false — so no value of x can ever satisfy this equation.
\[ 2x + 5 = 2x + 9 \;\Longrightarrow\; 5 = 9 \quad \text{false: no solution} \]
Why: The variable disappears entirely, leaving a statement about numbers only. Since five is not nine, that statement is false no matter what x is, so the equation has no solution. Geometrically the two sides describe parallel lines that never meet.
Comparison
Every linear equation ends in one of exactly three ways. Fill the blanks.
Comparison matrix
| what you end up with | how many solutions | example |
|---|---|---|
| x equals a number | exactly one | 3x = 12 gives x = 4 |
| a false number statement | none | 5 = 9 |
| a true number statement | infinitely many | 7 = 7, so every x works |
The last two feel like something has gone wrong. Nothing has — the equation is simply telling you something about all numbers at once.
Pattern
Every equation in this chapter yields to the same five moves, in this order.
Checking against the original matters: if you made an error on line two, checking against line three would happily confirm it.
Picture it
Each side of a linear equation is a line. Where they cross is the solution — and that picture explains all three cases at once.
Figure (svg): Three coordinate planes showing two lines crossing once, two parallel lines never crossing, and two identical lines overlapping everywhere
So the strange cases are not algebra glitches. They are the two ways straight lines can fail to cross exactly once.
Section
Section 3.5
Concept
Fractions are not harder, they are just slower. Multiply every term by the least common denominator and they vanish.
\[ \frac{x}{2} + \frac{x}{3} = 5 \quad \Longrightarrow \quad 6 \cdot \frac{x}{2} + 6 \cdot \frac{x}{3} = 6 \cdot 5 \]
\[ 3x + 2x = 30 \]
Figure (svg): Each term of an equation being multiplied by six, with the denominators cancelling to leave whole numbers
Worked example
Solve the equation below.
\[ \frac{x}{2} + \frac{x}{3} = 5 \]
Find the least common denominator
Why: The denominators are two and three, so the smallest number both divide into is six.
Multiply every term on both sides by 6
Why: Every term, including the plain five. Missing one is the only way this method fails.
\[ 3x + 2x = 30 \]
Combine like terms, then divide
Why: Three x plus two x is five x, and dividing both sides by five isolates x.
\[ 5x = 30 \;\Longrightarrow\; x = 6 \]
Figure (svg): A bar of length x split as one half plus one third, with the remaining sixth shown, totalling five when x is six
Verify: substitute 6 into the original equation
Why: Six over two is three and six over three is two, and three plus two is five, matching the right side exactly.
Constraint
The tool you just learned has been taken away.
Discussion prompt
Solve x over 2 plus x over 3 equals 5 without multiplying through. What do you have to do instead, and which route would you choose in a test?
Hint: Can you combine the two fractions on the left into a single fraction?
Answer:
You would have to add the fractions first, using a common denominator inside the expression: x over two plus x over three is five x over six.
\[ \frac{5x}{6} = 5 \;\Longrightarrow\; 5x = 30 \;\Longrightarrow\; x = 6 \]
Same answer, and honestly about the same length here. Clearing denominators wins decisively once there are three or more fractions, or when they sit on both sides.
Matching
Before solving, decide what to multiply through by. That single choice removes most of the difficulty.
Match the pairs
Why: In each case you want the smallest number every denominator divides into. Fifths and tenths both divide into ten, so ten is enough and there is no need to use fifty. Decimals follow the same logic in disguise: one decimal place means denominators of ten.
Section
Section 3.6
Concept
A decimal equation is a fraction equation in disguise. Multiply every term by a power of ten to clear it.
\[ 0.4x + 1.2 = 3.6 \quad \Longrightarrow \quad 4x + 12 = 36 \]
Count the most decimal places in any single term — that decides whether you multiply by ten, a hundred, or a thousand.
Worked example
Solve the equation below.
\[ 0.4x + 1.2 = 3.6 \]
Count the decimal places
Why: Every term has one decimal place, so multiplying by ten will clear them all.
Multiply every term by 10
Why: Both sides, every term. Now the arithmetic is whole numbers and far less error-prone.
\[ 4x + 12 = 36 \]
Solve the whole-number equation
Why: Subtract twelve from both sides, then divide by four.
\[ 4x = 24 \;\Longrightarrow\; x = 6 \]
Figure (svg): Three decimal terms each being multiplied by ten to become whole numbers
Verify: substitute 6 into the original decimal equation
Why: Zero point four times six is two point four, and two point four plus one point two is three point six, matching the right side.
Estimation
A quick estimate is the cheapest error check there is.
\[ 0.9x = 44.1 \]
Predict first
Roughly what size is x?
Correct: About 50 — the exact value is 49.
\[ x = \frac{44.1}{0.9} = 49 \]
Why: Dividing by zero point nine is almost the same as dividing by one, so x should be a little larger than 44. That lands near 49, and any answer near 5 or 400 has a misplaced decimal point. The exact division gives 44.1 divided by 0.9, which is 49.
Sorting
Sort each equation by the multiplier that clears every decimal in one move.
Sort into buckets
What should you multiply through by?
Section
Section 3.7
Concept
A formula relates several quantities. Making one letter the subject uses exactly the same moves as solving for x.
\[ A = \ell w \quad \Longrightarrow \quad w = \frac{A}{\ell} \]
Figure (svg): A rectangle labelled with area A, length l and width w, with the formula rearranged beside it
Worked example
The perimeter of a rectangle is given below. Solve it for the width.
\[ P = 2\ell + 2w \]
Subtract the term that does not contain w
Why: Two times the length is added to the w term, so subtracting it from both sides is the first inverse.
\[ P - 2\ell = 2w \]
Divide both sides by the coefficient of w
Why: The coefficient is two, exactly as it would be if it were a plain number equation.
\[ w = \frac{P - 2\ell}{2} \]
Figure (svg): A rectangle perimeter broken into two lengths and two widths, with the two lengths being removed to leave two widths
Verify: test the rearranged formula on a known rectangle
Why: A rectangle of length 5 and width 3 has perimeter 16. The formula gives 16 minus 10, all over 2, which is 3 — the width we started with, so the rearrangement is correct.
Explain it to yourself
The moves are identical to solving for x, yet formulas feel worse. Say why.
Discussion prompt
What actually makes solving for w harder than solving for x, given the steps are the same?
Hint: What normally tells you a solving step worked?
Answer:
Because you cannot simplify as you go. With numbers, three plus four collapses to seven and the expression gets shorter; with letters, P minus two l just sits there, so the line never shrinks and there is no feedback that you are making progress.
The fix is to trust the procedure rather than the shrinking. Name the target letter first, then ask the same two questions: what is added to it, and what is multiplying it.
Reverse engineer
Here is a formula already rearranged. Reconstruct what it looked like before.
Fill in the blanks
h = \fracbh/2___ \;\text___\; A = ___
Why: The area of a triangle is half the base times the height. Multiplying both sides by two gives 2A equals b times h, and dividing by b makes h the subject. Working backwards like this is a fast way to check that a rearrangement did not lose a factor.
Missing information
A problem reads: a rectangle has perimeter 30 centimetres. Find its area.
Discussion prompt
Why can this not be answered, and what one extra fact would fix it?
Hint: Try to find two different rectangles that both have perimeter 30.
Answer:
Perimeter alone does not fix a rectangle's shape. A 14 by 1 rectangle and an 8 by 7 rectangle both have perimeter 30, but their areas are 14 and 56 — wildly different.
You need one more piece: either the length, the width, or a relationship between them such as the length being twice the width. With any one of those, the perimeter formula becomes a solvable equation in a single unknown.
Section
Section 3.8
Concept
A ratio compares two quantities by dividing. A rate is a ratio whose two quantities have different units.
Figure (svg): A bar model showing a ratio of two to three split into five equal parts, with the parts labelled
unit rate — A rate written with a denominator of one, such as 55 miles per hour or 9 dollars per hour. It is the most useful form because it says what happens per single unit.
Worked example
Two people share 45 dollars in the ratio two to three. How much does each get?
Count the total number of parts
Why: Two parts plus three parts is five equal parts in the whole.
Write the equation for one part
Why: If one part is worth p dollars, then five parts is the whole 45.
\[ 5p = 45 \;\Longrightarrow\; p = 9 \]
Scale each share by its own number of parts
Why: The first person has two parts and the second has three.
\[ 2p = 18 \qquad 3p = 27 \]
Figure (svg): A bar of forty five dollars split into five parts of nine, with two parts shaded for one person and three for the other
Verify: add the two shares and reduce the ratio
Why: Eighteen plus twenty-seven is forty-five, the correct total, and eighteen to twenty-seven divides down to two to three, the correct ratio.
Real world
A 12-ounce bottle costs 1.80 dollars. A 20-ounce bottle costs 2.80 dollars.
Discussion prompt
Work out both unit rates and say which is better value. Why is the unit rate the right tool here?
Hint: What would you have to make the same before the two prices could be compared fairly?
Answer:
The small bottle is 1.80 divided by 12, which is 15 cents per ounce. The large is 2.80 divided by 20, which is 14 cents per ounce, so the large bottle is better value.
The unit rate is the right tool because the two bottles are different sizes: comparing the prices directly compares two different things. Dividing to a per ounce figure puts them on the same footing, which is the entire job of a rate.
Prediction
A recipe uses 3 cups of flour for every 2 cups of milk. You use 12 cups of flour.
Predict first
How much milk do you need?
Correct: 8 cups — the flour was multiplied by four, so the milk must be too.
\[ \frac{3}{2} = \frac{12}{8} \]
Why: Twelve is four times three, so the whole recipe has been scaled by four and the milk goes from two cups to eight. The tempting wrong answer is eleven, which comes from adding nine to both quantities instead of multiplying: ratios are preserved by multiplying, never by adding the same amount to both parts.
Invariant
Watch a ratio being scaled and name the quantity that never moves.
Step through it
One thing is identical in every frame. What is it?
The ratio is the invariant. That is precisely why you may scale a recipe up or down and it still tastes the same.
Section
Section 3.9
Concept
A percent is a ratio with 100 as its denominator. Every percent problem is one equation: part equals percent times whole.
Figure (svg): A percent bar showing 25 percent of 80 as a quarter of a bar, marked as 20
\[ \text{part} = \text{percent} \cdot \text{whole} \]
Worked example
Fifteen is 25 percent of what number?
Identify the part, the percent and the whole
Why: Fifteen is the part, twenty-five percent is the percent, and the whole is what is unknown. Labelling first is what stops the numbers being multiplied in the wrong role.
Write the percent as a decimal and build the equation
Why: Twenty-five percent is zero point two five.
\[ 15 = 0.25w \]
Divide both sides by the coefficient
Why: The whole is being multiplied by zero point two five, so divide by it.
\[ w = \frac{15}{0.25} = 60 \]
Figure (svg): A bar showing fifteen as a quarter of a bar whose whole is sixty
Verify: take 25 percent of the answer
Why: A quarter of sixty is fifteen, which is the part we were given, so sixty is the correct whole.
Discrimination
Three questions look alike and need different equations. Sort them by what is unknown.
Sort into buckets
In each question, which quantity is missing?
Check
One equation, four plausible answers. Paper first.
Check your understanding
Solve 4(x - 3) = 2x + 6.
Answer: A
Why: Distributing gives 4x minus 12 equals 2x plus 6. Subtracting 2x from both sides gives 2x minus 12 equals 6, then adding 12 gives 2x equals 18, so x is 9. Checking: the left is 4 times 6, which is 24, and the right is 18 plus 6, which is also 24.
Check
Same care, different topic.
Check your understanding
A jacket costs 60 dollars after a 25 percent discount. What was the original price?
Answer: A
Why: After a 25 percent discount you pay 75 percent of the original, so 60 equals 0.75 times the original. Dividing 60 by 0.75 gives 80. Checking: 25 percent of 80 is 20, and 80 minus 20 is 60.
Commit first
Answer, then rate your confidence.
\[ \text{Solve } 3(x - 2) = 3x - 6 \]
Predict first
How many solutions does this equation have?
Correct: Infinitely many — the two sides are the same expression.
\[ 3(x - 2) = 3x - 6 \;\Longrightarrow\; -6 = -6 \quad \text{always true} \]
Why: Distributing the left gives 3x minus 6, which is identical to the right side. Subtracting 3x from both sides leaves negative six equals negative six, a true statement with no x in it, so every number is a solution. This is an identity rather than an equation to be solved.
Exit ticket
Thirty seconds that will save you an hour later.
Predict first
Which of these would you least want to meet in a test right now?
Correct: Whatever you picked is the one to drill first.
Why: All four use the same five-move recipe, so the gap is almost never conceptual — it is usually one specific move, such as dividing by a negative or remembering to multiply the constant term too. Naming which one turns an hour of vague revision into ten targeted problems.
Connect it up
Draw it once and the whole chapter collapses into a single procedure.
Draw it
Draw the five-move recipe down the middle of the page. Off each move, branch to the situations that trigger it: brackets, fractions, decimals, variables on both sides, formulas, ratios, percents. Mark the two moves you most often forget.
Every problem in this chapter is that same spine with a different decoration hanging off it.
Recap
You can now find the number the letter was hiding, in every form Algebra 1 will present it.
| if you remember one thing | it should be |
|---|---|
| about legality | whatever you do, do it to both sides and to every term |
| about order | undo the last operation first |
| about strange answers | no variable left means no solution or every solution |
| about checking | substitute into the original, never into your own later line |
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