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 or that they have the same value, for example,, or. o To solve an equation means to find all the 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, has exactly two solutions,. 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. 3. A contradiction is never true and has no solution. For example, is not true for any value of.

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 Although other types of equations may have more than one solution, every linear equation has exactly one solution. 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. Continued on next slide…

8. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Solving Linear Equations To solve a linear equation (cont.): 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.

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

10. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 2: Solving Linear Equations No Solution Solve: Since is a contradiction, this equation has no solution.

11. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 3: Solving Linear Equations Solve:

12. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 4: Solving Linear Equations All real numbers, a.b. Solve:

13. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 5: Solving Linear Equations All real numbers, Solve:

14. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 6: Solving Linear Equations No Solution Solve:

15. 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 it’s negative. 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. means or

16. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Solving Absolute Value Equations Caution! The apparent solutions obtained by the previous method may not solve the original equation! Absolute value equations are one class of equations in which it is very important to check your final answer in the original equation. An apparent solution that does not solve the original problem is called an extraneous solution.

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

18. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 2: Solving Absolute Value Equations Because of the two absolute value quantities, the one original equation could potentially lead to four linear equations. But two of them are equivalent to the other two. Thus, we are left with two equations to solve. or Solve:

19. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 3: Absolute Value Equations or Solve:

20. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example 3: Absolute Value Equations (cont.) However, if we check the solutions, we find that No Solution