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Step 10: Pre-algebra

Linear equations in one variable

Step 10 · Pre-algebra topics

Solve linear equations with the variable on both sides, distribute and collect like terms first, clear fractions, and recognize equations with no solution or infinitely many.

Variables on both sides

When x appears on both sides, first gather the x terms on one side and the plain numbers on the other. Then solve as before.

5x + 3 = 2x + 15. Take 2 x's off each pan to leave 3x + 3 = 15.

Example: x on both sides

Solve 5x + 3 = 2x + 15.

  1. Subtract 2x from both sides: 3x + 3 = 15.
  2. Subtract 3: 3x = 12.
  3. Divide by 3: x = 4.
  4. Check: 5 × 4 + 3 = 23 and 2 × 4 + 15 = 23.

Answerx = 4

Example: Negative coefficients

Solve 7 − 2x = 3x − 8.

  1. Add 2x to both sides: 7 = 5x − 8.
  2. Add 8: 15 = 5x.
  3. Divide by 5: x = 3.

Answerx = 3

Distributing and collecting terms

Parentheses come off first, using the distributive property: a(b + c) = ab + ac. Then collect like terms: x terms together and plain numbers together.

Example: Distributing

Solve 3(x + 4) = 2(x − 1) + 19.

  1. Distribute: 3x + 12 = 2x − 2 + 19.
  2. Collect: 3x + 12 = 2x + 17.
  3. Subtract 2x, then 12: x = 5.

Answerx = 5

Example: A minus outside

Simplify 8 − 2(x − 3).

  1. −2 times x is −2x, and −2 times −3 is +6.
  2. So it is 8 − 2x + 6, which is 14 − 2x.

Answer14 − 2x

Example: Fractions

Solve x2+x3=10\frac{x}{2} + \frac{x}{3} = 10.

  1. Multiply every term by 6: 3x + 2x = 60.
  2. 5x = 60, so x = 12.

Answerx = 12

One, none or infinitely many solutions

A linear equation usually has exactly one solution. But when the x terms cancel, two other things can happen.

After simplifyingExampleSolutions
x = a number3x + 1 = 7 gives x = 2one
a false statement2x + 1 = 2x + 5 gives 1 = 5none
a true statement2(x + 3) = 2x + 6 gives 6 = 6infinitely many

Example: No solution

Solve 4x − 3 = 4(x + 1).

  1. Distribute: 4x − 3 = 4x + 4.
  2. Subtract 4x: −3 = 4, which is false.

AnswerNo solution

Example: Every number works

Solve 3(2x − 1) = 6x − 3.

  1. Distribute: 6x − 3 = 6x − 3.
  2. The two sides are the same for every x.

AnswerInfinitely many solutions

Try it yourself

Work each one on paper first, then open the answer to check.

  1. Solve 6x − 5 = 2x + 11.

    Show answer

    x = 4

  2. Solve 9 − x = 2x + 3.

    Show answer

    x = 2

  3. Solve 2(x + 5) = 3x − 4.

    Show answer

    x = 14

  4. Simplify 10 − 3(x + 2).

    Show answer

    4 − 3x

  5. Solve x4+x2=9\frac{x}{4} + \frac{x}{2} = 9.

    Show answer

    x = 12

  6. How many solutions does 5x + 2 = 5x − 7 have?

    Show answer

    None: it simplifies to 2 = −7.

  7. How many solutions does 4(x − 1) = 4x − 4 have?

    Show answer

    Infinitely many: the two sides are the same.

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