Sullivan Algebra and Trigonometry: Section 1.1

Objectives of this Section• Solve an Equation in One Variable

• Solve a Linear Equation

• Solve Equations That Lead to Linear Equations

The set of all values of a variable that make an equation a true statement are called solutions, or roots, of the equation.

Example: The statement x + 5 = 9 is true when x = 4 and false for any other choice of x. Thus, 4 is a solution of the equation. We also say that 4 satisfies the equation x + 5 = 9.

We solve equations by creating a series of equivalent equations, that is equations that have precisely the same solution set.

Example: The equations 3x – 4 = 5x and 2x = -4 are equivalent equations, since the solution set for both equation is {-2}.

Procedures that Result in Equivalent Equations

1. Interchange the two sides of the equation.

2. Simplify the sides of the equation by combining like terms, eliminating parenthesis, and so on.

3. Add or subtract the same expression on both sides of the equation.

4. Multiply or divide both sides of the equation by the same nonzero expression.5. If one side of the equation is 0 and the other side can be factored, then set each factor equal to 0.

Solve:

Example

( ) ( )3 3 3 32x x x− = −

( ) ( )3 3 3 32x x x− = −

3 9 3 92 2x x x− = −

− =−9 9xx =1

Example

Solve:2

14 8

1x

x x−+ =

−

( ) ( ) ( )x xx

xx

x−−

+ − =−

−1 21

4 1 81

1

2 4 4 8x x+ − =6 4 8x− =

6 12x =x =2

ExampleSolve the equation:

( )( ) ( )( )x x x x + = + +3 1 2 4

x x x x2 22 3 6 8− − = + +− − = +2 3 6 8x x

− = +2 6 11x x− =8 11xx = −11

8

A linear equation in one variable is equivalent to an equation of the form:

ax + b = 0

Note that the previous example,

( )( ) ( )( )x x x x + = + +3 1 2 4Is an example of a linear equation, since it is equivalent to:

8x + 11 = 0

Both equations have the same solution set: {-11/8}

Example

Solve the equation: 2(x - 3) = 5x - 9

2x - 6 = 5x - 92x = 5x - 9 + 6

-3x = - 3

x = 1