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# Equations and inequalities: Strategy for solving equations

## Strategy for Solving Equations

This chapter is all about solving different types of equations for one or two variables. In general, we want to get the unknown variable alone on the left hand side of the equation with all the constants on the right hand side of the equation. For example, in the equation x-1=0x-1=0


, we want to be able to write the equation as x=1x=1.

As we saw in rearranging equations, an equation is like a set of weighing scales that must always be balanced. When we solve equations, we need to keep in mind that what is done to one side must be done to the other.

### Method: Rearranging Equations

You can add, subtract, multiply or divide both sides of an equation by any number you want, as long as you always do it to both sides.

For example, in the equation x+5-1=-6x+5-1=-6


, we want to get xx alone on the left hand side of the equation. This means we need to subtract 5 and add 1 on the left hand side. However, because we need to keep the equation balanced, we also need to subtract 5 and add 1 on the right hand side.

x + 5 - 1 = - 6 x + 5 - 5 - 1 + 1 = - 6 - 5 + 1 x + 0 + 0 = - 11 + 1 x = - 10 x + 5 - 1 = - 6 x + 5 - 5 - 1 + 1 = - 6 - 5 + 1 x + 0 + 0 = - 11 + 1 x = - 10
(1)

In another example, 23x=823x=8, we must divide by 2 and multiply by 3 on the left hand side in order to get xx alone. However, in order to keep the equation balanced, we must also divide by 2 and multiply by 3 on the right hand side.

2 3 x = 8 2 3 x Ã· 2 Ã— 3 = 8 Ã· 2 Ã— 3 2 2 Ã— 3 3 Ã— x = 8 Ã— 3 2 1 Ã— 1 Ã— x = 12 x = 12 2 3 x = 8 2 3 x Ã· 2 Ã— 3 = 8 Ã· 2 Ã— 3 2 2 Ã— 3 3 Ã— x = 8 Ã— 3 2 1 Ã— 1 Ã— x = 12 x = 12
(2)

These are the basic rules to apply when simplifying any equation. In most cases, these rules have to be applied more than once, before we have the unknown variable on the left hand side of the equation.

#### Tip:

The following must also be kept in mind:
1. Division by 0 is undefined.
2. If xy=0xy=0, then x=0x=0 and yâ‰ 0yâ‰ 0, because division by 0 is undefined.

We are now ready to solve some equations!

#### Investigation : 4 = 3 ??

In the following, identify what is wrong.

x = 2 4 x - 8 = 3 x - 6 4 ( x - 2 ) = 3 ( x - 2 ) 4 ( x - 2 ) ( x - 2 ) = 3 ( x - 2 ) ( x - 2 ) 4 = 3 x = 2 4 x - 8 = 3 x - 6 4 ( x - 2 ) = 3 ( x - 2 ) 4 ( x - 2 ) ( x - 2 ) = 3 ( x - 2 ) ( x - 2 ) 4 = 3
(3)

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