Variables on Both Sides

Step 1: Let's Learn

Read it, or press Listen and follow the words.

Try it first

Try x = 2, x = 3 and x = 4 in 5x + 3 = 2x + 12. Which one makes both sides the same number?

Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.

Watch the idea first — 56 seconds. Then read on, and try it yourself in the next step.

When x appears on both sides, gather every x term on one side and every number on the other. Then it is the equation you already know how to solve.

Which side

Move the smaller x term. Subtracting 2x from 5x leaves 3x, a positive coefficient. Subtracting 5x from 2x leaves −3x, which works but invites a sign slip. Both roads give x = 3.

Whatever you do, do to both

The equation is a balance. Taking 2x from one pan without the other tips it, and the line you write is no longer true. Every move goes in the margin so a reader can see it was done to both sides.

Brackets first

4(x − 1) = 2x + 6 has a bracket. Distribute before gathering: 4x − 4 = 2x + 6, then 2x − 4 = 6, then 2x = 10, so x = 5. Brackets, then collect, then gather, then undo.

The letter can end up on the right

x + 8 = 4x − 1: subtract x from both sides, 8 = 3x − 1. Add 1: 9 = 3x. Divide by 3: 3 = x. Gathering on the right kept every coefficient positive. x = 3 either way round.

Always check

Substitute the answer into the original equation. 5(3) + 3 = 18 and 2(3) + 12 = 18. Both sides give the same number, so the answer is right. Different numbers mean a mistake somewhere above.

Step 2: Try It Yourself

Tap and try it out.

Press Next step for each move. The orange note in the margin is what was done to both sides.

Solve 4(x − 1) = 2x + 6

14(x − 1) = 2x + 6the equation as given

Step 0 of 5

x tiles on both pans. Take x tiles off both until one pan has none, then the units, then share.

5x + 3 = 2x + 12

5x + 3 balances 2x + 12. Take the same off both pans and they stay level, because the same was done to each side.

x tiles on the left

Units on the left

Units on the right

x tiles on the right

Step 3: In Real Life

Two car rentals

One company charges 40 a day plus 0.20 a mile, another 25 plus 0.35 a mile. When are they equal? 40 + 0.2m = 25 + 0.35m: 100 miles. Beyond that, the first is cheaper.

Step 4: Watch an Example

One step at a time.

Watch Dev Solve 7x − 4 = 3x + 16

x on both sides, and a negative number to move.

Step 1
Both sides carry an x term, and 3x is the smaller. Taking 3x from both sides leaves 4x − 4 = 16, with x on the left only.

Finish Maya's Solve of 9x − 2 = 5x + 10

Maya writes the first two lines. The last two are yours.

Step 1
Both sides carry an x term. 5x is the smaller, so taking 5x from both sides leaves 4x − 2 = 10.

Two Roads Through 6x + 2 = 4x + 14

Either x term can be taken from both sides. Leo takes the smaller one, Ana the larger. Watch what each road costs.

Step 1
Leo takes the smaller x term. Taking 4x from both sides leaves 2x + 2 = 14, a positive coefficient of x.

Step 5: Your Turn

Practice makes it stick.

Two Bike Rentals

Problem 1 of 2

Shop A rents a bike for $12 plus $3 an hour. Shop B charges $4 plus $5 an hour. After how many hours do the two shops cost the same?

hours

Two Gyms

Problem 2 of 2

Gym A costs $20 to join plus $5 a month. Gym B costs $35 to join plus $2 a month. After how many months have the two gyms cost the same?

months

Solve and Check

Take the smaller x term first. Write each move in the margin, then check in the original.

1 of 5

Solve 3x + 5 = x + 13.

2 of 5

Solve 8x = 3x + 35.

3 of 5

Solve 6x − 7 = 2x + 13.

4 of 5

Solve x + 18 = 4x.

5 of 5

Solve 2(x + 5) = 4x − 6.

Whose Step?

Each page has one line that does not follow from the line above it. Find the line, then fix it.

1 of 6

Priya solved 6x + 1 = 2x + 13, but her answer does not check. Which line has the slip?

Priya: solve 6x + 1 = 2x + 13

16x + 1 = 2x + 13
24x + 1 = 13− 2x
34x = 14− 1
Answerx = 3.5÷ 4

2 of 6

Fix Priya's line 3 and finish her solve. What is x?

3 of 6

Noor solved 3(x + 2) = 5x − 4, and her check gives 15 on one side and 11 on the other. Which line has the slip?

Noor: solve 3(x + 2) = 5x − 4

13(x + 2) = 5x − 4
23x + 2 = 5x − 4
32 = 2x − 4− 3x
46 = 2x+ 4
Answerx = 3÷ 2

4 of 6

Write Noor's line 2 correctly: 3(x + 2) = 5x − 4 with the bracket distributed in full.

5 of 6

Leo solved 2x + 9 = 5x − 6 by taking the larger x term. His answer does not check. Which line has the slip?

Leo: solve 2x + 9 = 5x − 6

12x + 9 = 5x − 6
23x + 9 = −6− 5x
33x = −15− 9
Answerx = −5÷ 3

6 of 6

Fix Leo's line 2 and finish his solve. What is x?

Pick the Move

No solving needed. Choose the first move, and the reason for it.

1 of 2

7x − 3 = 2x + 17. Which x term do you take from both sides first, and why?

2 of 2

x + 15 = 6x. Which first move, and why?

Same Solution?

Do not solve. For each equation, ask what was done to both sides.

1 of 2

5x + 3 = 2x + 12 has solution x = 3. Without solving, select every equation that has the same solution.

2 of 2

2(x + 4) = 6x − 12. Without solving, select every equation that has the same solution.

Step 6: Quick Check

Show what you know.

Question 1 of 1

9x + 2 = 4x + 22. What is x?

Ready for more?Optional. Past the lesson, for anyone who wants it.

Stretch 1 of 2

For which value of k does 5x + 3 = 2x + k have the solution x = 0?

Stretch 2 of 2

Sam wants an equation with x on both sides whose solution is x = −2. He starts with 4x + c = x + 5. What must c be?

What You Learned

Gather the x terms on one side and the numbers on the other.
Move the smaller x term so the coefficient stays positive.
Brackets first, then collect, then gather, then undo.
Check in the original: both sides must give the same number.

For the grown-up

The slip this lesson targets is a term carried across the equals sign without its sign changing, or taken from one side only. Before the next line is written, ask what was done to both sides; the margin note exists to make that a habit rather than a check.