Using Algebra Tiles to Solve One-Step and Two-Step Equations

Solving algebra tiles equations comes down to three balance-mat rules: whatever you do to one side, do to the other; a positive and a negative tile of the same kind cancel as a zero pair; and when only x-tiles are left on one side, split the other side into equal groups. For (x + 4 = 9), remove 4 unit tiles from each side to get (x = 5).

Prefer a scale model? Try the free Balance Algebra game, where x-boxes and unit weights sit on a balance and you solve one-step and two-step equations by doing the same thing to both sides.

Follow along on a real mat: Try it with our free Algebra Tiles tool. Its equation mat keeps both sides balanced and records each move in a step log.

The tiles you need

• x-tile: a long rectangle worth (x).
• Unit tile: a small square worth (1).

Positive tiles are usually yellow, green or white. Shaded (or red) tiles are negative, so a shaded unit tile is worth (-1) and a shaded x-tile is worth (-x).

An equation mat has two halves, one for each side of the equals sign. Think of it as a balance scale. These three rules do all the work:

1. Keep the mat balanced. Any tile you add to or remove from one side, you add to or remove from the other side.
2. Use zero pairs. One positive and one negative tile of the same kind are worth 0 together. You can remove a zero pair, or add one, without changing a side’s value.
3. Divide equally. When one side is only x-tiles, share the tiles on the other side equally among them. One share is the value of x.

Step by step: one-step equations with algebra tiles

Adding: x + 4 = 9

1. Build the equation: 1 x-tile and 4 unit tiles on the left, 9 unit tiles on the right.
2. To get the x-tile alone, remove 4 unit tiles from each side.
3. The left has 1 x-tile and the right has 5 unit tiles: (x = 5).
4. Check: (5 + 4 = 9).

Subtracting: x − 3 = 7

1. Build it: 1 x-tile and 3 negative (shaded) unit tiles on the left, 7 positive unit tiles on the right.
2. To clear the negative units, add 3 positive unit tiles to each side. On the left they form 3 zero pairs; remove them.
3. The left has 1 x-tile and the right has (7 + 3 = 10) unit tiles: (x = 10).
4. Check: (10 – 3 = 7).

Multiplying: 2x = 8 and 3x = 12

For (2x = 8), put 2 x-tiles on the left and 8 unit tiles on the right. Split the 8 units into 2 equal groups, one beside each x-tile. Each group has 4, so (x = 4). For (3x = 12), split 12 unit tiles into 3 groups of 4, so again (x = 4). Check: (2(4) = 8) and (3(4) = 12).

Two-step: 2x + 3 = 11

A two-step equation simply combines the moves: first clear the unit tiles from the x side, then divide.

1. Build it: 2 x-tiles and 3 unit tiles on the left, 11 unit tiles on the right.
2. Remove 3 unit tiles from each side: (2x = 8).
3. Split the 8 units into 2 equal groups: (x = 4).
4. Check: (2(4) + 3 = 11).
5. Try it. Open the free Algebra Tiles tool, pick the equation mat, enter (2x + 3 = 11) and make the same two moves. The step log will show (2x = 8) and then (x = 4).

The order matters. Clear the units before you divide; if you split the left side into groups while unit tiles are still there, the groups get messy fast.

More worked examples

Example 1: 2x − 5 = 3

The left has 2 x-tiles and 5 shaded unit tiles. The right has 3 positive unit tiles.

1. Add 5 positive unit tiles to each side. On the left they cancel the 5 negative units as zero pairs. The right now has 8 units: (2x = 8).
2. Split 8 units into 2 groups: (x = 4).

Check: (2(4) – 5 = 3).

Example 2: 3x + 8 = 11

1. Remove 8 unit tiles from each side: (3x = 3).
2. Split 3 units among 3 x-tiles: (x = 1).

Check: (3(1) + 8 = 11).

Example 3: a negative coefficient, −2x = 6

Here the left side has 2 shaded x-tiles and the right has 6 positive unit tiles. You cannot share positive units among negative x-tiles directly, so flip every tile on the mat. Flipping both sides is the same as multiplying both sides by (-1), so the mat stays balanced.

1. Flip all tiles: the left becomes 2 positive x-tiles and the right becomes 6 shaded unit tiles, so (2x = -6).
2. Split the 6 negative units into 2 groups of 3: (x = -3).

Check: (-2(-3) = 6).

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Common mistakes on the equation mat

• Changing only one side. Removing 4 units from the left of (x + 4 = 9) but not from the right gives (x = 9), which fails the check.
• Removing tiles that are not there. In (x – 3 = 7) there are no positive units on the left to take away. Add 3 positive units to both sides and cancel the zero pairs instead.
• Dividing before clearing units. In (2x + 3 = 11), clear the 3 units first. Dividing 11 by 2 straight away gives 5.5, which is not the answer.
• Forgetting to flip both sides. With (-2x = 6), flip the tiles on both sides, not just the x-tiles. Flipping only the left gives (2x = 6) and the wrong answer (x = 3).
• Reading a shaded tile as positive. Count shaded tiles as negative every time. One misread tile changes the answer.

Practice exercises with answers

Model each equation with tiles, solve, then check by substituting.

1. (x + 5 = 11)
2. (x – 4 = 6)
3. (3x = 15)
4. (2x = 14)
5. (x + 7 = 7)
6. (4x = 20)

Answers:

1. Remove 5 units from each side: (x = 6).
2. Add 4 positive units to each side and cancel the zero pairs: (x = 10).
3. Split 15 units into 3 groups: (x = 5).
4. Split 14 units into 2 groups: (x = 7).
5. Remove 7 units from each side. Nothing is left on the right, so (x = 0).
6. Split 20 units into 4 groups: (x = 5).

Ready for more? Try these two-step ones: (3x + 2 = 14) (answer (x = 4)) and (2x – 1 = 9) (answer (x = 5)).

Algebra tiles equations practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

How do you use algebra tiles to solve equations?

Build each side of the equation on a mat, with x-tiles for the variable and unit tiles for numbers, and shaded tiles for negatives. Make the same change to both sides until the x-tiles are alone, using zero pairs whenever a side does not have the tiles you need to remove. Then split the other side equally among the x-tiles. One share is x.

What are the steps to solve an equation with algebra tiles?

Four steps: model both sides, clear the unit tiles from the x side (remove them, or add opposites and cancel zero pairs), divide the other side into as many equal groups as there are x-tiles, and check the answer in the original equation. For (2x + 3 = 11), that is remove 3, split 8 into 2 groups, and (x = 4).

How do you solve one-step equations with algebra tiles?

Do the one move that leaves the x-tile alone. If units are added to x, remove the same number from both sides. If negative units are with x, add the same number of positive units to both sides. If there are several x-tiles and nothing else, split the other side into equal groups.

How do algebra tiles handle negative numbers?

Negative numbers are shaded (or red) tiles. A shaded tile and a matching positive tile cancel as a zero pair. If the x-tiles themselves are negative, as in (-2x = 6), flip every tile on both sides of the mat. That gives (2x = -6), so (x = -3).

What if there are x-tiles on both sides?

Then remove the same number of x-tiles from both sides first, so they stay on one side only, and continue as usual. Our lesson on solving equations with variables on both sides using algebra tiles works through (3x + 2 = x + 4) and balanced-scale problems.

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• Free Algebra Tiles virtual manipulative
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