An equation is a claim that two expressions are equal, and this deck starts there. From that idea it builds one-step equations, two-step equations, and equations with the variable on both sides, using the balance model throughout. It also covers the classic trap of operating on only one side, and every worked example ends with a verification step.
Subject: Algebra 1 · 30 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Objectives
By the end of this deck you can:
1. Explain what an equation actually claims, using the balance model.
2. Solve one-step and two-step equations by undoing operations on both sides.
3. Solve equations with the variable on both sides by collecting like terms first.
4. Catch the most common mistake - changing only one side - and verify every answer by substituting it back.
Picture it
Figure (svg): A balanced two-pan scale with x plus 7 in the left pan and 12 in the right pan
Discussion prompt
Read the picture before the words. What is this showing, and what is the one thing it is built to make obvious? Commit to an answer, then read on.
Hint: Name the parts, then say what changes between them — and if nothing changes, say what is being held still.
Answer:
An equation is a claim that two quantities are equal. The equals sign is a balanced scale, not a command to compute.
Concept
An equation is a claim that two quantities are equal. The equals sign is a balanced scale, not a command to compute.
\[ x + 7 = 12 \]
Figure (svg): A balanced two-pan scale with x plus 7 in the left pan and 12 in the right pan
solution — A value of the variable that makes the claim true - the number that keeps the scale level when you substitute it in.
Counterexample
Discussion prompt
An equation is a claim that two quantities are equal. The equals sign is a balanced scale, not a command to compute.
That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.
Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.
Intuition
Picture taking 7 grams off the left pan only. The scale tips - the claim is no longer true, and anything you conclude afterward is about a different, broken scale.
Take 7 grams off both pans and the scale stays level. The claim is still true - just simpler. Solving is nothing but simplifying a true claim until the variable stands alone.
That is the whole game: every legal move keeps both sides equal. There are no other rules to memorize.
Analogy
Discussion prompt
Explain Why 'do it to both sides'? by analogy to something with no Algebra 1 in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.
Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.
Answer:
Picture taking 7 grams off the left pan only. The scale tips - the claim is no longer true, and anything you conclude afterward is about a different, broken scale.
Fill the middle
Fill in the blanks
From One-step equation — finish the line. Write what belongs on the right of the equals sign before you look.
x + 7 - 7 = 12 - 7
Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. Addition is undone by subtraction, and changing both sides the same way keeps the balance true.
Worked example
Solve:
\[ x + 7 = 12 \]
Subtract 7 from both sides
Why: Addition is undone by subtraction, and changing both sides the same way keeps the balance true.
\[ x + 7 - 7 = 12 - 7 \]
\[ x = 5 \]
Verify by substituting 5 back into the original equation
Why: If both sides come out equal, the answer is certain - no guessing.
\[ 5 + 7 = 12 \checkmark \]
Picture it
Animation
Shows: Each line of the worked example "One-step equation", appearing one at a time.
The same working the example does, in the order a tutor would write it.
Takeaway: If both sides come out equal, the answer is certain - no guessing.
Concept
When the variable was multiplied and then shifted, undo the operations in reverse order: clear the added or subtracted number first, then divide away the coefficient.
\[ 3x - 4 = 11 \]
Think of unwrapping a present: the last layer wrapped on is the first layer taken off.
Explain it
Discussion prompt
Explain Two-step equations: undo in reverse order to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.
Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.
Answer:
When the variable was multiplied and then shifted, undo the operations in reverse order: clear the added or subtracted number first, then divide away the coefficient.
Ranking
Put in order
Put the moves of Two-step equation into the order they have to happen.
Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. The minus 4 was applied last, so it comes off first; adding 4 to both sides keeps the balance.
Worked example
Solve:
\[ 3x - 4 = 11 \]
Add 4 to both sides
Why: The minus 4 was applied last, so it comes off first; adding 4 to both sides keeps the balance.
\[ 3x = 15 \]
Divide both sides by 3
Why: Multiplication is undone by division; dividing both sides by the same nonzero number keeps the balance.
\[ x = 5 \]
Verify by substituting 5 back into the original equation
Why: Three times five is fifteen; fifteen minus four is eleven, which matches the right side.
\[ 3(5) - 4 = 15 - 4 = 11 \checkmark \]
Picture it
Animation
Shows: Each line of the worked example "Two-step equation", appearing one at a time.
The same working the example does, in the order a tutor would write it.
Takeaway: Three times five is fifteen; fifteen minus four is eleven, which matches the right side.
Anomaly
Predict first
A student writes this, and it looks reasonable:
Subtract 6 from the left side only
It is wrong. Say what breaks — and say it before you turn the page.
Correct: Feels like just 'erasing the 6' - but the right side was never touched, so the scale tipped.
Feels like just 'erasing the 6' - but the right side was never touched, so the scale tipped.
Why: Feels like just 'erasing the 6' - but the right side was never touched, so the scale tipped.
Trap
\[ 2x + 6 = 14 \]
Subtract 6 from the left side only
Why: Feels like just 'erasing the 6' - but the right side was never touched, so the scale tipped.
\[ 2x = 14 \]
Divide both sides by 2
Why: This move is fine - but it is operating on an already-broken claim.
\[ x = 7 \]
Check it
Why: Two times seven plus six is twenty, not fourteen. The check exposes the broken step.
\[ 2(7) + 6 = 20 \neq 14 \]
\[ 2x + 6 = 14 \]
Subtract 6 from BOTH sides
Why: The balance survives only when both pans change together.
\[ 2x = 8 \]
Divide both sides by 2
Why: Same legal move as before - now applied to a claim that is still true.
\[ x = 4 \]
Check it
Why: Two times four plus six is fourteen - both sides agree, so this answer is certain.
\[ 2(4) + 6 = 14 \checkmark \]
Notation
Annotate
From Trap: changing only one side — read this one piece at a time. What is each part doing?
On: \( 2(7) + 6 = 20 \neq 14 \)
Concept
Nothing new is needed: variable terms are quantities too, so you may add or subtract a variable term from both sides just like a number.
\[ 5x - 2 = 3x + 8 \]
Analogy
Discussion prompt
Explain Variables on both sides by analogy to something with no Algebra 1 in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.
Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.
Answer:
Nothing new is needed: variable terms are quantities too, so you may add or subtract a variable term from both sides just like a number.
Intuition
You get to choose. Most people collect the variable on the side that starts with more of it - here the left side - so the coefficient stays positive and there are fewer sign slips.
Either choice gives the same answer. The balance model does not care which pan you tidy first - it only cares that every move hits both pans.
Counterexample
Discussion prompt
You get to choose. Most people collect the variable on the side that starts with more of it - here the left side - so the coefficient stays positive and there are fewer sign slips.
That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.
Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.
Answer:
Either choice gives the same answer. The balance model does not care which pan you tidy first - it only cares that every move hits both pans.
Ranking
Put in order
Put the moves of Variable on both sides into the order they have to happen.
Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. Collect all variable terms on the left, where the coefficient is larger; both sides change together.
Worked example
Solve:
\[ 5x - 2 = 3x + 8 \]
Subtract 3x from both sides
Why: Collect all variable terms on the left, where the coefficient is larger; both sides change together.
\[ 2x - 2 = 8 \]
Add 2 to both sides
Why: Clear the constant from the variable's side; the balance is preserved.
\[ 2x = 10 \]
Divide both sides by 2
Why: Undo the coefficient last, leaving the variable alone.
\[ x = 5 \]
Verify by substituting 5 into BOTH original sides
Why: Left side: five times five minus two is twenty-three. Right side: three times five plus eight is twenty-three. Both sides agree.
\[ 5(5) - 2 = 23 \quad\text{and}\quad 3(5) + 8 = 23 \checkmark \]
Picture it
Animation
Shows: Each line of the worked example "Variable on both sides", appearing one at a time.
The same working the example does, in the order a tutor would write it.
Takeaway: Left side: five times five minus two is twenty-three. Right side: three times five plus eight is twenty-three. Both sides agree.
Explain it to yourself
Discussion prompt
In The procedure (works for every linear equation) this move is made:
5. Verify by substituting the answer into the ORIGINAL equation
Why is that legal? Name the rule or definition it rests on before you read on.
Hint: If you can only say "because that is what you do", the rule is the thing to go and find.
Answer:
Both sides must come out identical - this step costs ten seconds and catches every trap in this deck.
Pattern
1. Simplify each side on its own
Why: Distribute and combine like terms - no balance moves needed yet.
2. Collect all variable terms on one side
Why: Add or subtract a variable term from both sides; prefer the side with the larger coefficient.
3. Undo addition or subtraction on both sides
Why: Clear the constant away from the variable term.
4. Undo multiplication or division on both sides
Why: Divide (or multiply) both sides by the coefficient, leaving the variable alone.
5. Verify by substituting the answer into the ORIGINAL equation
Why: Both sides must come out identical - this step costs ten seconds and catches every trap in this deck.
Real world
Discussion prompt
Outside this lesson: where does Solving Linear Equations actually turn up? Name one concrete situation — a job, a piece of software someone ships, a decision somebody has to make — and say which part of The procedure (works for every linear equation) is doing the work in it.
Hint: Vague is the failure mode here. "Engineering" is not a situation; "deciding whether this build is fast enough to ship" is.
Answer:
An equation is a claim that two expressions are equal, and this deck starts there. From that idea it builds one-step equations, two-step equations, and equations with the variable on both sides, using the balance model throughout. It also covers the classic trap of operating on only one side, and every worked example ends with a verification step.
Check
Solve it on paper first - including the verify step - then pick your answer.
Check your understanding
Solve 4x + 9 = 21.
Answer: A
Why: Subtract 9 from both sides: 4x = 12. Divide both sides by 4: x = 3. Verify in the original equation: 4(3) + 9 = 12 + 9 = 21, which matches the right side exactly, so x = 3 is certain.
Connect it up
Draw it
One page, no notation unless you need it: draw how these connect — The procedure (works for every linear equation) · An equation is a balance · Why 'do it to both sides'? · Two-step equations: undo in reverse order · Variables on both sides. Put an arrow wherever one of them is what makes another possible, and label the arrow with why.
Recap
An equation is a balanced claim, and every legal move changes both sides the same way. You solved one-step, two-step, and both-sides equations with the same five-step procedure.
| To undo... | Do this to both sides |
|---|---|
| an added number | subtract it |
| a subtracted number | add it |
| a coefficient (multiplication) | divide by it |
| a variable term on the other side | subtract that variable term |
And always verify: substitute the answer back and watch both sides agree.
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