4. Solving Equations

Remember this kind of problem from primary school?

? + 5 = 7

We just needed to figure out what number went into the box to make it a true statement. Clearly, we need to replace the question mark with "2":

2 + 5 = 7

Solving equations using algebra is really no different. Instead of using a box, we use a letter to represent a number. Our task is to find the correct number (or sometimes there may be more than one number) that makes the equation true.

Sometimes we can "see" the right answer if it is simple (maybe we can just count up with our fingers, or whatever.) But when our equations become more complicated, we need a process to follow that will eventually give us the answer.

Our Process

  1. We are aiming to get x (or whatever letter the question uses) on the left hand side of the equals sign, by itself.
  2. We solve equations by balancing: whatever we do to one side of an equation, we must do the same to the other side. So if we add 4 to the left hand side, we must add 4 to the right hand side as well. If we multiply on the left side by 2, we multiply on the right side by 2 as well.

Example 1

Solve the equation

x − 6 = 10

Answer


Example 2

Solve 5x = 35

Answer


Example 3

Solve

3x/4 = 7

Answer


Example 4

Solve 5 − (x + 2) = 5x

Answer


Example 5

Solve 5x − 2(x − 5) = 4x

Answer


Example 6

In solving the equation

7 − (2 − x) = x + 2,

what conclusion can be made?

Answer




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