1. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra Section 2.1a: Linear Equations in One Variable

2. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Objectives o Equivalent equations and the meaning of solutions. o Solving linear equations. o Solving absolute value equations.

3. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Equations and the Meaning of Solutions o An equation is a statement that two algebraic expressions are equal. o To solve an equation means to find the solution(s): value(s) of the variable that make the equation true. o The set of all such values is called the solution set.

4. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Types of Equations There are three types of equations: 1. A conditional equation has a countable number of solutions. For example, x + 7 = 12 has one solution, 5. The solution set is {5}. 2. An identity is true for all real numbers and has an infinite number of solutions. For example, is true for all real number values of. The solution set is R. 3. A contradiction is never true and has no solution. For example, is false for any value of. The solution set is Ø.

5. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Linear Equations in One Variable A linear equation in one variable, such as the variable, is an equation that can be transformed into the form, where and are real numbers and. Such equations are also called first-degree equations, as appears to the first power.

6. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Linear Equations and Equivalent Equations o We solve linear equations by performing the same operations on both sides of the equation. o This results in simpler equivalent equations that are easier to solve and have the same solution.

7. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Solving Linear Equations To solve a linear equation (in x): 1. Simplify each side of the equation separately by removing any grouping symbols and combining like terms. 2. Add or subtract the same expression(s) on both sides of the equation in order to get the variable term(s) on one side and the constant term(s) on the other side of the equation and simplify. 3. Multiply or divide by the same nonzero quantity on both sides of the equation in order to get the numerical coefficient of the variable term to be one. 4. Check your answer by substitution in the original equation.

8. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example: Solving Linear Equations Step 1: Simplify Step 2: Add or Subtract Step 3: Multiply or Divide Solve:

9. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example: Solving Linear Equations The solution set is R. Solve:

10. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example: Solving Linear Equations The solution set is R. Solve:

11. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example: Solving Linear Equations No Solution Solve: The solution set is Ø.

12. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example: Solving Linear Equations Solve.

13. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Solving Absolute Value Equations The absolute value of any quantity is either the original quantity or its negative (opposite). This means that, in general, every occurrence of an absolute value term in an equation leads to two equations with the absolute value signs removed, if c > 0. Note: if c < 0, it has no solution. means or ax + b = -c

14. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example: Solving Absolute Value Equations Step 1: Rewrite the absolute value equation without absolute values. Step 2: Solve the two equations or 3x – 2 = -5 or3x = -3 or Solve:

15. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example: Absolute Value Equations Solve: |4x + 3| = -2 False, absolute value is never negative. No solution; the solution set is Ø. Solve: |6x – 2| = 0 6x – 2 = 0 6x = 2 x = ⅓ If |ax + b| = 0, then ax + b = 0.

16. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example: Absolute Value Equations Solve. |x – 4| = |2x + 1|