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LearnAlgebra

Equations with x on both sides

Move every x to one side and every number to the other, solve equations with fractions, and rearrange a formula.

Sometimes the unknown x appears on both sides of an equation, like 7x + 3 = 4x + 18. This happens all the time when you compare two choices. One gym costs $30 to join and then $5 a visit. Another costs nothing to join but $10 a visit. After how many visits do they cost the same? With x visits, the first costs 30 + 5x and the second costs 10x, so you need to solve 30 + 5x = 10x. Below you learn to solve equations like this, equations made of two fractions, and how to turn a formula around to find a different letter.

Bring the x terms together

The rule has not changed: whatever you do to one side, you do to the other. The new idea is that you can also take away an x term from both sides. Take away the smaller x term, so that the x terms are only on one side. After that, it is a normal two-step equation.

  1. Take away the smaller x term from both sides.
  2. Add or take away a number on both sides, so that the numbers are on the other side.
  3. Divide both sides by the number in front of x.
  4. Check: put your answer into BOTH sides of the first equation. They must give the same number.

Example 1

Solve 7x + 3 = 4x + 18.

  1. The smaller x term is 4x. Take 4x away from both sides:
  2. 3x + 3 = 18
  3. Take 3 away from both sides: 3x = 15
  4. Divide both sides by 3: x = 5
  5. Check: left side 7(5) + 3 = 38. Right side 4(5) + 18 = 38. They are equal.

Answer: x = 5

Example 2

Solve 5x − 8 = 3x + 6.

  1. Take 3x away from both sides: 2x − 8 = 6
  2. Add 8 to both sides: 2x = 14
  3. Divide both sides by 2: x = 7
  4. Check: left side 5(7) − 8 = 27. Right side 3(7) + 6 = 27. They are equal.

Answer: x = 7

Example 3

Gym A costs $30 to join and $5 a visit. Gym B costs $10 a visit and nothing to join. After how many visits do they cost the same?

  1. Let x be the number of visits. Then 30 + 5x = 10x.
  2. Take 5x away from both sides: 30 = 5x
  3. Divide both sides by 5: x = 6
  4. Check: Gym A costs 30 + 5(6) = 60 dollars. Gym B costs 10(6) = 60 dollars.

Answer: 6 visits (both cost $60)

Two fractions that are equal

An equation like x/4 = 15/20 says that two fractions are equal. An equation of this kind is called a proportion. There are two good ways to solve it.

Way 1: simplify the known fraction. Divide the top and the bottom of 15/20 by 5, and you get 3/4. Now the equation is x/4 = 3/4. Both fractions have 4 at the bottom, so the tops must be equal too: x = 3.

Way 2: cross-multiply. Multiply both sides by both bottom numbers. In practice this means: the top of each fraction times the bottom of the OTHER fraction. The two products are equal.

Remember

If a/b = c/d, then ad = bc.

Each top is multiplied by the other bottom. This is called cross-multiplying.

Example 4

Solve 6/x = 3/5.

  1. Cross-multiply: 6(5) = 3x
  2. 30 = 3x
  3. Divide both sides by 3: x = 10
  4. Check: 6/10 simplifies to 3/5. Correct.

Answer: x = 10

Example 5

Solve x/6 = 2/3.

  1. Cross-multiply: 3x = 6(2)
  2. 3x = 12
  3. Divide both sides by 3: x = 4
  4. Check: 4/6 simplifies to 2/3. Correct.

Answer: x = 4

Rearranging a formula

A formula is an equation with letters that has a use, like C = 2πr for the distance around a circle. "Solve for r" means: rearrange the formula so that r is alone on one side. You use exactly the same steps as before. The only difference is that the answer has letters in it.

Example 6

Solve y = 3x + 6 for x.

  1. Take 6 away from both sides: y − 6 = 3x
  2. Divide both sides by 3. The WHOLE left side is divided: (y − 6)/3 = x
  3. Write it the usual way round: x = (y − 6)/3

Answer: x = (y − 6)/3

Example 7

Solve C = 2πr for r.

  1. 2πr means 2π times r.
  2. Divide both sides by 2π: C/2π = r
  3. So r = C/2π

Answer: r = C/2π

Try it yourself

  1. 1.Solve 6x − 5 = 2x + 11.

    Show the answer

    x = 4

    Take 2x away: 4x − 5 = 11. Add 5: 4x = 16. Divide by 4: x = 4. Both sides give 19.

  2. 2.Solve 3x + 14 = 8x − 1.

    Show the answer

    x = 3

    Take 3x away: 14 = 5x − 1. Add 1: 15 = 5x. So x = 3. Both sides give 23.

  3. 3.Solve x/9 = 10/15.

    Show the answer

    x = 6

    10/15 simplifies to 2/3, which is 6/9. Or cross-multiply: 15x = 90, so x = 6.

  4. 4.Solve 8/x = 2/7.

    Show the answer

    x = 28

    Cross-multiply: 8(7) = 2x, so 56 = 2x and x = 28.

  5. 5.Solve y = 5x − 4 for x.

    Show the answer

    x = (y + 4)/5

    Add 4 to both sides: y + 4 = 5x. Divide the whole side by 5.

  6. 6.Solve d = st for t.

    Show the answer

    t = d/s

    st means s times t. Divide both sides by s.

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