Section 6.2Solving Linear Equations

Math in Our World

Learning Objectives

Decide if a number is a solution of an equation. Identify linear equations. Solve general linear equations. Solve linear equations containing fractions. Solve formulas for one specific variable. Determine if an equation is an identity or a

contradiction.

Equations

An equation is a statement that two algebraic expressions are equal.

A solution of an equation is a value of the variable that makes the equation a true statement when substituted into the equation. Solving an equation means finding every solution of the equation. We call the set of all solutions the solution set, or simply the solution of an equation.

For example, x = 2 is one solution of the equation x2 – 4 = 0, because (2)2 – 4 = 0 is a true statement. But x = 2 is not the solution, because x = – 2 is a solution as well. The solution set is actually {– 2, 2}.

Expressions vs. Equations

Note the difference between the two; equations contain an equal sign and expressions do not.

EXAMPLE 1 Identifying Solutions of an Equation

Determine if the given value is a solution of the equation.

(a) 4(x – 1) = 8; x = 2

(b) x + 7 = 2x – 1; x = 8

(c) 2y2 = 200; y = – 10

Linear Equations

A linear equation does not break the following rules:*variables do not contain exponents greater than 1*variables are not square rooted*variables do not appear in the denominator of a fraction*variables are not multiplied together

EXAMPLE 2 Identifying Linear Equations

Determine which of the equations below are linear equations.

Solving Linear Equations

One-step Equations*Addition/Subtraction Properties

*Multiplication/Division Properties

EXAMPLE 3 Solving Linear Equations

Solve each equation using the Addition and Subtraction Property, and check your answer.

(a) x – 5 = 9

(b) y + 30 = 110

EXAMPLE 4 Solving Linear Equations

Solve each equation using the Multiplication and Division Property, and check your answer.

Solving Linear EquationsMulti-step Equations

Procedures for Solving Linear EquationsStep 1 Simplify the expressions on both sides of the equation by distributing and combining like terms.Step 2 Get the term with the variable by itself using the addition or subtraction propertiesStep 3 Use the multiplication or division properties to solve for the variable.

EXAMPLE 5 Solving a Linear Equation

Solve the equation 5x + 9 = 29.

EXAMPLE 6 Solving a Linear Equation

Solve the equation 6x – 10 = 4x + 8.

EXAMPLE 7 Solving a Linear Equation

Solve the equation 3(2x + 5) – 10 = 3x – 10.

Solving EquationsContaining Fractions

There’s a simple procedure that will turn any equation with fractions into one with no fractions at all. You just need to find the Least CommonDenominator of all fractions that appear in the equation, and multiply every single term on each side of the equation by the LCD.

If there are any fractions left after doing so, you made a mistake!

EXAMPLE 8 Solving Linear Equations Containing Fractions

Solve the equation:

EXAMPLE 9 Solving Linear Equations Containing Fractions

Solve the equation: 422

5

3

32

xxx

Solve formulas for one specific variable.

EXAMPLE 10 Solving a Formula in Electronics for One Variable

Solve the formula

EXAMPLE 11 Finding a Formula for Temperature in Celsius

The formula F = 95 C + 32 gives the Fahrenheit equivalent for a temperature in Celsius.

Transform this into a formula for calculating the Celsius temperature C.

Contradictions and Identities

A contradiction is an equation with no solution.

An identity is an equation that is true for any value of the variable for which both sides are defined.

When you solve an equation that is an identity, the final equation will be a statement that is always true. In a contradiction the final equation will be a statement that is false.

EXAMPLE 12 Recognizing Identities and Contradictions

Indicate whether the equation is an identity or a contradiction, and give the solution set.

(a) 3(x – 6) + 2x = 5x – 18

(b) 6x – 4 + 2x = 8x – 10

Classwork

p. 286-287: 7-71 eoo, 75, 77, 83