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8TH GRADE MATH · FREE LESSON

How to solve equations with variables on both sides

When x appears on both sides, you can still solve by preserving equality. Collect the variable terms, isolate the remaining variable, and check the result in the original equation.

Start the lesson
BEFORE YOU BEGIN

A quick starting check.

  • Solve a two-step equation by applying the same operation to both sides.
  • Combine like terms such as 5x - 2x.
  • Substitute a number for a variable and evaluate the expression.
UNDERSTAND THE IDEA

Build the method, one step at a time.

Collect variable terms with an equal operation

In 5x+2=2x+145x + 2 = 2x + 14, subtracting 2x from both sides removes the variable term from the right. The equation becomes 3x+2=143x + 2 = 14. You have not moved a term by a special rule; you have performed the same subtraction on both sides.

You could subtract 5x from both sides instead and get 2=−3x+142 = -3x + 14. That route is valid too. Choosing the smaller coefficient often keeps the remaining coefficient positive, which can make the arithmetic easier.

Finish as a two-step equation

Once the variable is on one side, undo addition or subtraction and then undo multiplication or division. Keep track of signs when combining terms. In 5x−2x5x - 2x, the result is 3x because five copies minus two copies leaves three copies.

Always substitute into both sides of the original equation. The solution must make those two values equal. An answer that only fits a mistakenly simplified line does not solve the original problem.

If x disappears, read the statement that remains

Sometimes all the variable terms cancel. From 2x+3=2x+72x + 3 = 2x + 7, subtracting 2x leaves 3=73 = 7. That statement is false for every value of x, so the equation has no solution.

From 2x+3=2x+32x + 3 = 2x + 3, the same subtraction leaves 3=33 = 3. The statement is true for every real value of x, so every real number is a solution. Neither case means x is zero. These two cases are an extension to help you interpret a cancelled variable, not a reason to divide by zero.

SEE IT WORK

Two worked examples.

EXAMPLE 1

Solve 5x + 2 = 2x + 14

  1. Subtract 2x from both sides: 3x+2=143x + 2 = 14.
  2. Subtract 2 from both sides: 3x=123x = 12. Divide by 3 to get x=4x = 4.
  3. Evaluate both original sides: 5(4)+2=225(4) + 2 = 22 and 2(4)+14=222(4) + 14 = 22.

Check your reasoning: Both expressions equal 22 when x is 4. The solution is x=4x = 4; 22 is the shared value of the expressions, not the value of x.

EXAMPLE 2

Solve 2x + 9 = 5x - 6

  1. Subtract 2x from both sides: 9=3x−69 = 3x - 6.
  2. Add 6 to both sides: 15=3x15 = 3x. Divide by 3 to get x=5x = 5.
  3. Evaluate the originals: 2(5)+9=192(5) + 9 = 19 and 5(5)−6=195(5) - 6 = 19.

Check your reasoning: The variable can end up on the right while solving. The statements 5=x5 = x and x=5x = 5 mean the same thing.

YOUR TURN

Try it before you reveal it.

Write down your thinking. Open a hint if you get stuck, then compare your method with the explanation.

Practice 1

Solve 4x+1=2x+94x + 1 = 2x + 9.

Show a hint

Subtract 2x from both sides to leave a two-step equation.

Show the answer and explanation

x=4x = 4

Subtracting 2x gives 2x+1=92x + 1 = 9. Subtract 1 and divide by 2 to get x=4x = 4. Both original sides then equal 17.

Practice 2

Does 3x+4=3x+103x + 4 = 3x + 10 have one solution, no solution or every real number as a solution?

Show a hint

Subtract 3x from both sides and read what remains.

Show the answer and explanation

No solution

The equation reduces to 4=104 = 10, which is false regardless of x. No value of x can make the original expressions equal.

Practice 3

What are the solutions of 6x−2=6x−26x - 2 = 6x - 2?

Show a hint

Both expressions are identical. Try subtracting 6x from each side.

Show the answer and explanation

Every real number

Subtracting 6x leaves −2=−2-2 = -2, a true statement. Any real value substituted for x gives the same result on both sides.

KEEP THIS IDEA

What to take with you.

Use equal operations to collect the variable terms. If a variable remains, solve and substitute; if it cancels, decide whether the remaining statement is always true or always false.

Try explaining one example without looking at the steps. If a step is hard to explain, revisit that part before moving on.

A PROMPT TO TRY IN STUDY

Give me an equation with variables on both sides. Ask which variable term I want to subtract from both sides, then help me check each transformation without giving away the solution.

Eligible students ages 13 and older with Study access can sign in and use this prompt for more practice. Check AI explanations against your class materials.

Continue in Study

Written by FutureSteps for study and review. This lesson expands a topic in our Texas starter collection. Course reference: Texas TEKS · Grade 8 Mathematics (§111.28(b)(8)(C)). These references do not imply endorsement, complete standard coverage or assessed mastery. Follow your school’s materials and course sequence.

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