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**LIAL HORNSBY SCHNEIDER**

COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER

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**1.1 Linear Equations Basic Terminology of Equations**

Solving Linear Equations Identities Solving for a Specified Variable

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**Equations An equation is a statement that two expressions are equal.**

x + 2 = x = 5x + 6x x2 – 2x – 1 = 0 To solve an equation means to find all numbers that make the equation true. The numbers are called solutions or roots of the equation. A number that is a solution of an equation is said to satisfy the equation, and the solutions of an equation make up its solution set. Equations with the same solution set are equivalent equations.

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**Addition and Multiplication Properties of Equality**

For real numbers a, b, and c: If a = b, then a + c = b + c. That is, the same number may be added to both sides of an equation without changing the solution set.

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**Addition and Multiplication Properties of Equality**

For real numbers a, b, and c: If a = b and c ≠ 0, then ac = bc. That is, both sides of an equation may be multiplied by the same nonzero number without changing the solution set.

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**Linear Equation in One Variable**

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

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Linear Equations A linear equation is also called a first-degree equation since the greatest degree of the variable is one. Linear equations Non-linear equations

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**Solve Solution: Example 1 SOLVING A LINEAR EQUATION**

Be careful with signs. Distributive Property Combine terms. Add x to each side. Combine terms. Add 12 to each side. Combine terms. Divide each side by 7.

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**A check of the solution is recommended.**

SOLVING A LINEAR EQUATION Example 1 Check: Original equation ? Let x = 2. A check of the solution is recommended. ? True

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**Solve Solution: Example 2**

CLEARING FRACTIONS BEFORE SOLVING A LINEAR EQUATION Example 2 Solve Solution: Multiply by 12, the LCD of the fractions. Distribute the 12 to all terms within parentheses! Distributive property Distributive property Combine terms.

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**Solve Solution: Example 2**

CLEARING FRACTIONS BEFORE SOLVING A LINEAR EQUATION Example 2 Solve Solution: Combine terms. Subtract 3t; subtract 16. Divide by 11.

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**Check: Example 2 CLEARING FRACTIONS BEFORE SOLVING A LINEAR EQUATION**

? Let t = − 4. ? True

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**Identities, Conditional Equations, and Contradictions**

An equation satisfied by every number that is a meaningful replacement for the variable is called an identity.

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**Identities, Conditional Equations, and Contradictions**

An equation satisfied by some numbers but not others, such as 2x =4, is called a conditional equation.

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**Identities, Conditional Equations, and Contradictions**

An equation that has no solution, such as x = x +1, is called a contradiction.

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**IDENTIFYING TYPES OF EQUATIONS**

Example 3 Decide whether this equation is an identity, a conditional equation, or a contradiction. a. Solution: Distributive property Combine terms Subtract x and add 8.

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**IDENTIFYING TYPES OF EQUATIONS**

Example 3 Decide whether this equation is an identity, a conditional equation, or a contradiction. a. Solution: Subtract x and add 8. When a true statement such as 0 = 0 results, the equation is an identity, and the solution set is {all real numbers}.

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**This is a conditional equation, and its solution set is {3}.**

IDENTIFYING TYPES OF EQUATIONS Example 3 Decide whether this equation is an identity, a conditional equation, or a contradiction. b. Solution: Add 4. Divide by 5. This is a conditional equation, and its solution set is {3}.

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**IDENTIFYING TYPES OF EQUATIONS**

Example 3 Decide whether this equation is an identity, a conditional equation, or a contradiction. c. Solution: Distributive property Subtract 9x. When a false statement such as − 3 = 7 results, the equation is a contradiction, and the solution set is the empty set or null set, symbolized by .

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**Identifying Linear Equations as Identities, Conditional Equations, or Contradictions**

If solving a linear equation leads to a true statement such as 0 = 0, the equation is an identity. Its solution set is {all real numbers}. If solving a linear equation leads to a single solution such as x = 3, the equation is conditional. Its solution set consists of a single element. If solving a linear equation leads to a false statement such as − 3 = 7, then the equation is a contradiction. Its solution set is .

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**Solving for a Specified Variable (Literal Equations)**

The solutions of a problem sometimes requires use of a formula and simple interest is an example of this… I is the variable for simple interest t is the variable for years P is the variable for dollars r is the variable for annual interest rate

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**Goal: Isolate t on one side.**

SOLVING FOR A SPECIFIED VARIABLE Example 4 Solve for t. a. Solution: Goal: Isolate t on one side. Divide both sides by Pr. or

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**Solving for a Specified Variable (Literal Equations)**

This formula gives the future or maturity value A of P dollars invested for t years at an annual interest rate r. A is the variable for future or maturity value t is the variable for years P is the variable for dollars r is the variable for annual simple interest rate

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**Goal: Isolate P, the specified variable.**

SOLVING FOR A SPECIFIED VARIABLE Example 4 Solve for P. b. Goal: Isolate P, the specified variable. Solution: Divide by 1 + rt. or

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**Solve for x. c. Solution: Example 4 SOLVING FOR A SPECIFIED VARIABLE**

Distributive property Isolate the x terms on one side. Combine terms. Divide by 2.

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**APPLYING THE SIMPLE INTEREST FORMULA**

Example 5 Latoya Johnson borrowed $5240 for new furniture. She will pay it off in 11 months at an annual simple interest rate of 4.5%. How much interest will she pay? Solution: r =.045 P = 5240 t =11/12 (year)

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**She will pay $216.15 interest on her purchase.**

APPLYING THE SIMPLE INTEREST FORMULA Example 5 Latoya Johnson borrowed $5240 for new furniture. She will pay it off in 11 months at an annual simple interest rate of 4.5%. How much interest will she pay? Solution: She will pay $ interest on her purchase.

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