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# Solving Equations

## Solving Equations with ther Variable on Both Sides

To solve an equation that has the variable on both sides, use the properties of equality to write an equivalent equation that has the variable on only one side. Then solve. When you solve equations that contain grouping symbols, you may need to use the distributive property to remove the grouping symbols. Some equations may have no solution because there is no value of the variable that will result in a true equation. For example, x + 1 = x + 2 has no solution; it cannot be true. An equation that is true for every value of the variable is called an identity . For example, x + x = 2 x is true for every value of x.

Example

Solve 3( x - 2) = 4 x + 5.

Solution

First use the distributive property to remove the parentheses.

3x - 6 = 4x + 5

Next, collect all the terms with x on one side of the equal sign by subtracting 3x from each side.

3x - 6 - 3x = 4x + 5 - 3x

 - 6 = x + 5 Add like terms. - 6 - 5 = x + 5 - 5 Subtract 5 from each side. - 11 = x

## Solving Equations and Formulas

Some equations contain more than one variable. To solve an equation or formula for a specific variable, you need to get that variable by itself on one side of the equation.

When you divide by a variable in an equation, remember that division by 0 is undefined. When you use a formula, you may need to use dimensional analysis, which is the process of carrying units throughout a computation.

Example

Solve the formula d = rt for t .

Solution

The variable t has been multiplied by r, so divide each side by r to isolate t.

Thus , where r 0.

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