How to Solve Equations with Variables on Both Sides Using Algebra Tiles

To solve equations using algebra tiles when x appears on both sides, build each side on an equation mat, remove the same number of x-tiles from both sides so the x-tiles sit on one side only, clear the unit tiles by removing them or by adding zero pairs, and then split what is left into equal groups. For (3x + 2 = x + 4), that gives (x = 1).

For a hands-on way to practice equations with variables on both sides, play our free algebra balance scale game: World 4 asks you to take x-boxes off both pans first, then finish the two-step equation.

Want to move the tiles yourself while you read? Try it with our free Algebra Tiles tool. Its equation mat keeps both sides balanced and writes every move in a step log.

The tiles you need

Equations with a variable on each side only need two kinds of tiles:

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

Each tile has a positive side and a negative side. In our pictures positive tiles are plain (white or orange), and shaded (or red) tiles are negative. A positive tile and a negative tile of the same kind form a zero pair: together they are worth 0, so you can add or remove a zero pair on either side at any time without changing the equation.

The equation mat has a line down the middle that stands for the equals sign. The one rule that keeps everything honest: whatever you do to one side, you do to the other. If you are new to the mat, start with one-step and two-step equations with algebra tiles and come back here.

Step by step: solve 3x + 2 = x + 4 with algebra tiles

Here is the equation we will model: three x-tiles and two unit tiles on the left are balanced with one x-tile and four unit tiles on the right, so (3x + 2 = x + 4). Every tile is positive.

1. Build both sides. Put 3 x-tiles and 2 unit tiles on the left side of the mat. Put 1 x-tile and 4 unit tiles on the right side.

   

2. Get the x-tiles on one side. The right side has fewer x-tiles (1), so remove 1 x-tile from each side. The left keeps 2 x-tiles and the right has none: (2x + 2 = 4). This keeps the remaining x-tiles positive.
3. Clear the unit tiles from the x side. Remove 2 unit tiles from each side. Now (2x = 2).
4. Split into equal groups. There are 2 x-tiles, so arrange the right side into 2 equal groups. Each x-tile lines up with 1 unit tile, so (x = 1).
5. Check. Put 1 back in for every x-tile in the original equation: left (3(1) + 2 = 5) and right (1 + 4 = 5). Both sides are worth 5, so the answer is right.
6. Try it. Open the free Algebra Tiles tool, choose the equation mat, enter (3x + 2 = x + 4) and make the same three moves. The step log should read the same equations you see above.

More worked examples

Example 1: −x + (−1) = 3x + (−5)

The left side has 1 negative x-tile and 1 negative unit tile. The right side has 3 positive x-tiles and 5 negative unit tiles.

1. The left side has fewer x-tiles (it has a negative one). Add 1 positive x-tile to each side. On the left it makes a zero pair with the negative x-tile, so remove the pair. The right now has 4 x-tiles: (-1 = 4x + (-5)).
2. To clear the 5 negative units on the right, add 5 positive unit tiles to each side. On the right they form 5 zero pairs. On the left, 1 negative unit and 5 positive units leave 4 positive units after you remove 1 zero pair: (4 = 4x).
3. Split the 4 unit tiles into 4 groups, one per x-tile: (x = 1).

Check: left (-1 + (-1) = -2), right (3 + (-5) = -2). Both sides match.

Example 2: a balanced scale with 4 x-tiles and 8 unit tiles on one side and 5 x-tiles and 4 unit tiles on the other

The equation is (4x + 8 = 5x + 4).

1. The left side has fewer x-tiles, so remove 4 x-tiles from each side: (8 = x + 4).
2. Remove 4 unit tiles from each side: (4 = x).

Check: (4(4) + 8 = 24) and (5(4) + 4 = 24). The scale balances, so (x = 4).

Example 3: 6 x-tiles and 3 unit tiles on one side, 5 x-tiles and 9 unit tiles on the other

The equation is (6x + 3 = 5x + 9). Remove 5 x-tiles from each side to get (x + 3 = 9), then remove 3 unit tiles from each side: (x = 6). Check: (6(6) + 3 = 39) and (5(6) + 9 = 39).

Example 4: an answer that is negative, 2x + 7 = 4x + 11

1. Remove 2 x-tiles from each side (the left has fewer): (7 = 2x + 11).
2. You cannot take 11 unit tiles away from a side that has only 7. Instead, add 11 negative unit tiles to each side. On the right they cancel the 11 positive units. On the left, 7 positive and 11 negative units make 7 zero pairs and leave 4 negative units: (-4 = 2x).
3. Split the 4 negative units into 2 groups. Each x-tile matches 2 negative units, so (x = -2).

Check: (2(-2) + 7 = 3) and (4(-2) + 11 = 3). A negative answer means each x-tile lines up with shaded tiles.

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Common mistakes with variables on both sides

• Removing tiles from one side only. If you take an x-tile off the left, take one off the right too. Otherwise the mat stops being balanced and the answer changes.
• Removing x-tiles by the larger count. In (4x + 8 = 5x + 4), removing 5 x-tiles from each side leaves a negative x-tile on the left. It works, but you must flip every tile at the end. Remove from the side with fewer x-tiles and the x-tiles you keep stay positive.
• “Removing” tiles that are not there. When a side has fewer units than you need to remove, add the opposite tiles to both sides and cancel zero pairs, as in Example 4.
• Splitting only part of a side. When you divide into groups, every tile on the other side must go into a group. If 6 units will not split evenly among 4 x-tiles, the answer is a fraction, not a mistake.
• Skipping the check. Substituting the answer into the original equation catches a lost tile.

Practice exercises with answers

Model each equation on the mat (or in the tool), solve, and check.

1. (5x + 1 = 2x + 7)
2. (4x + 3 = x + 12)
3. (2x + 9 = 5x + 3)
4. (3x + (-4) = x + 2)
5. (x + 8 = 3x + 12)
6. (-2x + 3 = x + (-6))

Answers:

1. Remove 2 x-tiles from each side: (3x + 1 = 7). Remove 1 unit: (3x = 6), so (x = 2).
2. Remove 1 x-tile: (3x + 3 = 12). Remove 3 units: (3x = 9), so (x = 3).
3. Remove 2 x-tiles: (9 = 3x + 3). Remove 3 units: (6 = 3x), so (x = 2).
4. Remove 1 x-tile: (2x + (-4) = 2). Add 4 positive units to each side: (2x = 6), so (x = 3).
5. Remove 1 x-tile: (8 = 2x + 12). Add 12 negative units to each side: (-4 = 2x), so (x = -2).
6. Add 2 positive x-tiles to each side to cancel the negative ones: (3 = 3x + (-6)). Add 6 positive units: (9 = 3x), so (x = 3).

Equations with variables on both sides 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 solve equations using algebra tiles?

Model each side of the equation on a mat, using x-tiles for the variable and unit tiles for numbers, with shaded tiles for negatives. Then do the same move to both sides: remove matching tiles or add zero pairs until the x-tiles are alone on one side. Finally, split the other side into as many equal groups as there are x-tiles. One group is the value of x.

How do you solve −x + (−1) = 3x + (−5) with algebra tiles?

Add one positive x-tile to each side, which cancels the negative x-tile on the left and leaves (-1 = 4x + (-5)). Add 5 positive unit tiles to each side to cancel the negative units on the right, leaving (4 = 4x). Split the 4 units among the 4 x-tiles, so (x = 1). Both sides then equal (-2).

How do you model 3x + 2 = x + 4 with algebra tiles?

Place 3 x-tiles and 2 unit tiles on the left side of the mat and 1 x-tile and 4 unit tiles on the right. To solve, remove 1 x-tile from each side, then 2 unit tiles from each side. You are left with 2 x-tiles balancing 2 unit tiles, so (x = 1).

What do you do when there are x-tiles on both sides?

Remove the same number of x-tiles from both sides until one side has none. Take away as many as the side with fewer x-tiles has; the x-tiles left over are then positive. If that side has negative x-tiles, add positive x-tiles to both sides instead so the zero pairs cancel.

When do algebra tiles stop being practical?

Tiles work best with small whole numbers. With coefficients like 25, decimals, or answers that are fractions (for example (4x = 6), which gives (x = 1.5)), the model becomes slow or cannot show the answer with whole tiles. At that point, write the same moves in symbols: subtract from both sides, then divide.

Related algebra tiles lessons

• Free Algebra Tiles virtual manipulative
• Using Algebra Tiles to Solve One-Step and Two-Step Equations
• Use Algebra Tiles to Identify Equivalent Expressions
• Adding and Subtracting Polynomials Using Algebra Tiles
• How to Multiply Two Binomials Using Algebra Tiles
• How to Factor Quadratics Using Algebra Tiles
• Multi-step equations
• Using diagrams to model and solve equations