# A Guide to the GRE/Working with Equations

## Working with Equations

An equation can be altered in any manner so long both sides are equally affected.

An equation is a statement that two quantities are equal to each other. Anything can be done to an equation so long as it is done to both sides.

x = 2y It is true that x is equal to 2 times y.

100x = 200y Multiplying both sides by 100 yields another true statement.

x + 50 = 2y + 50 Adding 50 to both sides also yields another true statement.

If there is a variable on both sides of an equation, consolidate it on one side.

3x - 12 = x + 2

``` - 2	   -2 		Subtract 2 from both sides
```

3x - 14 = x

```  -x	  -x		Subtract x from both sides
```

2x - 14 = 0

``` + 14    + 14		Add 14 to both sides
```

2x = 14

```÷ 2      ÷ 2		Divide both sides by 2
```

x = 7

In an inequality (such as x < y), the same rules of equations apply, except that when multiplying both sides of an inequality by a negative number, the inequality sign must be flipped around. For example, if a > b, then -a < -b.

### Practice

Solve for the variable in the following equations.

1. 6x - 4 = 1 + x

2. 2a = 3. 3h - 17 = h + 4

1. x = 1

6x - 4 = 1 + x Take the initial equation.

6x = 5 + x Add 4 to both sides.

5x = 5 Subtract x from both sides.

x = 1 Divide both sides by 5.

2. a =

2a = Take the initial equation.

2a2 = 16 Multiply both sides by a.

a2 = 8 Divide both sides by 2.

a == Take the square root of both sides.

3. h = 10.5

3h - 17 = h + 4 Take the initial equation.

3h = h + 21 Add 17 to both sides.

2h = 21 Subtract h from both sides.

h = 10.5 Divide both sides by 2.