Solving Equations I

Definitions:

Equation => a mathematical statement that consists of two expressions joined by an equal sign, =.

False statements

1 + 2 = 4

True statements

 1 + 2 = 3

Conditionally true statements x + 2 = 3 (conditional equation)
If the value of x is 1, this equation is true. Otherwise, this equation is false.

Linear equation in one variable: ax + b = c, where a, b, and c are real numbers. Where a ≠ 0, and x is a variable that
represents an unknown quantity. In a linear equation, the variable always has exponent 1.
A linear equations can not have a variable under a radical or in the denominator.
Linear equations can only have 1 variable.

Examples:

NOTICE:

Expression	Equation
Never contains an equal sign	always contains an equal sign
Simplify an expression	solve an equation
Ex: 4x + 2x to get 6x	4x + 2x = 6 to get x = 1

A solution of an equation is a number that, when substituted for the variable, makes the equation true.
The solution is said to satisfy the equation.

Is x = -3 a solution of the equation 3x + 10 = 4 + x?

Is p = 7 a solution of the equation 5(p – 6) = -5?

Principles of Equality

Equivalent equations are equations that have the same solution.
2x + 6 = 14 and 2x = 8
x = 4
 

Addition Principle of Equality
If a = b, then	If x = 5, then x + 2 = 5 + 2 check:
Subtraction Principle of Equality
If a = b, then	If x = 5, then x – 2 = 5 - 2 check:
Multiplication Principle of Equality
If a = b, then	If x = 5, then check:

Combining Principles to Solve Linear Equations

Concept 2: Solving Equations II

Solving Linear Equations that contain Fractions

(show 2 ways)

Solving Linear Equations that have Variables on both sides

Solving Linear Equations that contain Parentheses

General Procedure for Solving a Linear Equation: pg. 152

1. If fractions are present, multiply both sides of the equation by the LCD of all the fractions.
2. Simplify each side of the equation.
3. Use addition and subtraction to get all terms with the variable on one side and all constant terms
on the other side of the equation. Then, simplify each side of the equation.
4. Divide both sides of the equation by the coefficient of the variable.
5. Check your solution in the original equation.

These problems have exactly one solution.

These problems have no solution

These problems have every value of the variable as a solution. They are called an identity.

Review: When you solve a linear equation, if the simplified equation is:

A false statement, such as 8 = 1, the equation has no solution.
A true statement, such as 12 = 12, the equation is an identity.
Any value of the variable is a solution.
A true statement, such as x = 4, there is only one solution.

FORMULAS:

d = rt, solve for r

ax + b = c, solve for x

, for h
, for F

8x – y = 5, for x