1.3 Solve Linear Equations

Equation

A statement that two expressions are equal

Linear equation

A linear equation in one variable is an equation that can be written in the form ax + b = 0, where a and b are constants and a ¹ 0.

Solution

A number is a solution of an equation in one variable if substituting the number for the variable results in a true statement.

Equivalent equations

Two equations are equivalent equations if they have the same solution(s).

TRANSFORMATIONS THAT PRODUCE EQUIVALENT EQUATIONS
Addition Property of Equality	Add the same number to each side.	If a = b, then a + c = b + c.
Subtraction Property of Equality	Subtract the same number from each side.	If a = b, then a - c = b - c.
Multiplication Property of Equality	Multiply each side by the same nonzero number.	If a = b and c ¹ 0, then _a · c_ = _b · c_
Division Property of Equality	Divide each side by the same nonzero number.	If a = b and c ¹ 0, then a ¸ c = b ¸ c.

Example 1

Solve an equation with a variable on one side

Solve

Solution

	Write original equation.
_18_	Subtract _15_ from each side.
x	Multiply each side by , the reciprocal of
x = _24_	Simplify.
The solution is _24_.

Your Notes

Example 2

Solve an equation using the distributive property

3(3x - 2) = 7(4x + 3) - 31x	Original equation
_9x_- 6 = _28x_ + 21 - 31x	Distributive property
9x - 6 = -_3x_ + 21	Combine like terms.
_12x_ - 6 = 21	Add _3x_ to each side.
12x_ = _27_	Add _6_ to each side.
x =	Divide each side by _12_ and simplify.
The solution is