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Solving simple equations

Solve equations like 2x + 5 = 17 by doing the same thing to both sides, one step at a time, until x is alone.

An equation says that two things are equal. In x + 7 = 15, the left side and the right side are the same number, and the job is to find the x that makes this true. People solve equations every day without the letters: "I paid 15 dollars, the drink was 7, so the sandwich was 8." Solving an equation is the same thinking, written down one line at a time, so it still works when the numbers get harder. The method is the same for every equation in this lesson: get x alone on one side.

The one rule

Think of an equation as a balance with the same weight in both pans. If you take something out of one pan, the balance tips, unless you take the same out of the other pan. An equation works the same way.

Remember

Whatever you do to one side, do the same to the other side.

Add, take away, multiply or divide: always both sides, and always the whole side.

To get x alone, you undo what is being done to it. Each operation has an opposite that undoes it.

The equation hasUndo it byExample
x + 7taking away 7x + 7 = 15 gives x = 15 − 7 = 8
x − 9adding 9x − 9 = 4 gives x = 4 + 9 = 13
3xdividing by 33x = 21 gives x = 21/3 = 7
x/4multiplying by 4x/4 = 5 gives x = 5(4) = 20

One-step equations

In x + 7 = 15, a jump of 7 from x lands on 15. Go back 7 from 15 and you find x = 8.

Example 1

Solve x + 12 = 30.

  1. x has 12 added to it. Take 12 away from both sides.
  2. x + 12 − 12 = 30 − 12
  3. x = 18
  4. Check: 18 + 12 = 30. Correct.

Answer: x = 18

Example 2

Solve x − 6 = 13.

  1. x has 6 taken away. Add 6 to both sides.
  2. x − 6 + 6 = 13 + 6
  3. x = 19
  4. Check: 19 − 6 = 13. Correct.

Answer: x = 19

Example 3

Solve 4x = 36.

  1. 4x means 4 times x. Divide both sides by 4.
  2. 4x/4 = 36/4
  3. x = 9
  4. Check: 4(9) = 36. Correct.

Answer: x = 9

Example 4

Solve x/6 = 8.

  1. x is divided by 6. Multiply both sides by 6.
  2. x = 8(6)
  3. x = 48
  4. Check: 48/6 = 8. Correct.

Answer: x = 48

Two-step equations

In 2x + 5 = 17, two things happen to x: it is multiplied by 2, and then 5 is added. To undo them, go backwards. Undo the adding or taking away first. Then undo the multiplying.

  1. Add or take away a number on both sides, so that only the x term is left on its side.
  2. Divide both sides by the number in front of x.
  3. Check your answer in the first equation.

Example 5

Solve 5x − 3 = 27.

  1. Add 3 to both sides: 5x − 3 + 3 = 27 + 3
  2. 5x = 30
  3. Divide both sides by 5: x = 30/5
  4. x = 6
  5. Check: 5(6) − 3 = 30 − 3 = 27. Correct.

Answer: x = 6

Example 6

Solve 7x + 2 = 30.

  1. Take 2 away from both sides: 7x = 28
  2. Divide both sides by 7: x = 28/7
  3. x = 4
  4. Check: 7(4) + 2 = 28 + 2 = 30. Correct.

Answer: x = 4

Equations with a fraction

Sometimes x is divided by a number and then something is added, like x/3 + 2 = 6. Use the same order: first the adding or taking away, then undo the division by multiplying.

Example 7

Solve x/3 + 2 = 6.

  1. Take 2 away from both sides: x/3 = 4
  2. Multiply both sides by 3: x = 4(3)
  3. x = 12
  4. Check: 12/3 + 2 = 4 + 2 = 6. Correct.

Answer: x = 12

Example 8

Solve x/2 − 6 = 3.

  1. Add 6 to both sides: x/2 = 9
  2. Multiply both sides by 2: x = 9(2)
  3. x = 18
  4. Check: 18/2 − 6 = 9 − 6 = 3. Correct.

Answer: x = 18

Finding a whole expression

Some questions ask for an expression, not for x. For example: "If 3x + 2 = 20, what is 6x + 4?" You could find x first. But look at the two expressions. Every number in 6x + 4 is double the number in 3x + 2. So multiply both sides of the equation by 2.

Example 9

If 3x + 2 = 20, what is 6x + 4?

  1. Multiply both sides by 2: 2(3x + 2) = 2(20)
  2. Multiply every term: 6x + 4 = 40
  3. Check the long way: 3x = 18, so x = 6, and 6(6) + 4 = 36 + 4 = 40.

Answer: 40

Example 10

If 5x = 12, what is 10x + 1?

  1. 10x is double 5x: 10x = 2(12) = 24
  2. Add 1: 10x + 1 = 24 + 1 = 25
  3. Check the long way: x = 12/5 = 2.4, and 10(2.4) + 1 = 24 + 1 = 25.

Answer: 25

Try it yourself

  1. 1.Solve x − 14 = 9.

    Show the answer

    x = 23

    Add 14 to both sides: x = 9 + 14 = 23.

  2. 2.Solve 9x = 63.

    Show the answer

    x = 7

    Divide both sides by 9: x = 63/9 = 7.

  3. 3.Solve x/7 = 6.

    Show the answer

    x = 42

    Multiply both sides by 7: x = 6(7) = 42.

  4. 4.Solve 3x − 4 = 23.

    Show the answer

    x = 9

    Add 4 to both sides: 3x = 27. Divide by 3: x = 9.

  5. 5.Solve x/5 + 4 = 10.

    Show the answer

    x = 30

    Take 4 away from both sides: x/5 = 6. Multiply by 5: x = 30.

  6. 6.If 2x + 3 = 11, what is 6x + 9?

    Show the answer

    33

    6x + 9 is three times 2x + 3, so it is 3(11) = 33. Check: x = 4, and 6(4) + 9 = 33.

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