Linear Equations

Equations that can be written in the form ax + b = 0 where a and b are real numbers, with a0, are linear equations. Examples oflinear equations include 5y + 9 = 16 and 8x = 4

The following properties are used to solve linear equations.

PROPERTIES OF EQUALITY

For all real numbers a, b, and c:

1. If a = b then a + c = b + c Addition property of equality (The same number may be addedto both sides of an equation.)

2. If a = b then ac = bc Multiplication property of equality (Both sides of an equation may be multiplied by the same number.)

Solving Linear Equations

EXAMPLE

(a) If x -2 = 3 then x = 2 + 3 = 5 Addition property of equality

(b) If x/2=3 then x = 2·3 = 6 Multiplication property of equality

The following example shows how these properties are used to solve lineare quations. Of course, the solutions should always be checked by substitution inthe original equation.

EXAMPLE

Solve 2x - 5 + 8 = 3x + 2(2-3x)

Solution

2x - 5 + 8 = 3x + 4-6x Distributive property

2x + 3 = -3x + 4 Combine like terms

5x + 3 = 4 Add 3x to both sides

5x = 1 Add -3 to both sides

Multiply both sides by .

Check by substituting in the original equation. The left side becomes 2(1/5)-5+8 and the right side becomes 3(1/5)+2(2-3(1/5)). Verify that both of these expressions simplify to 17/5.