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hendricks_intermediate_algebra_1e_ch1_3

“������������������������������������������������������������������������������������������������” �������������������������������������� ������������������������������������������������������������������ SECTION 2.1 Solving Linear Equations In Chapter 1, we learned how to evaluate algebraic expressions and to simplify algebraic expressions. We will now turn our attention to a fundamental skill of algebra, solving equations. The first few sections of this chapter will focus on linear equations. We will learn how to solve equations and inequalities using the addition and multiplication properties. Real-life applications of these equations and inequalities, along with absolute value equations and inequalities, will also be examined. In Section 2.3, we will determine how much money needs to be invested at 5% annual interest to have $4000 in an account after 3 yr. To find this amount of money requires us to solve the equation, 4000 = P(1 + 0.05)3. This is an example of a linear equation. This section will provide the skills we need to solve such an equation. Linear Equations Now that we know how to work with algebraic expressions, we will learn how to solve algebraic equations. Specifically, our focus will be on linear equations. Some examples of linear equations are shown. x + 5 = 9 2m - 3 = 4(m + 2) 3 2 y - 6 = 2 3 y + 1 Note that in each of these equations, the variable has an exponent of 1. Recall that x = x1. We can define a linear equation as follows. Definition: A linear equation in one variable is an equation that can be written in the form ax + b = c, where a, b, and c are real numbers and a ≠ 0. The exponent of the variable in a linear equation is 1. Linear equations are also called first-degree equations. Some examples of equations that are not linear are x2 - x = 6 and 4a3 + 1 = 9. Note that the largest exponent of the variable in these equations is greater than 1, which contradicts the definition of a linear equation. Our ultimate goal is to solve a linear equation. A solution of a linear equation in one variable is the value of the variable that satisfies the equation. That is, when the variable is replaced by this value, the resulting statement is true. The solution set of an equation is the set of all numbers that make the equation true. For example, if 3 is a solution of a linear equation, then the solution set is written as 536. Once we obtain a proposed solution of an equation, it is important to verify that this value is, in fact, the solution of the equation. For example, we know that 10 is a solution of the linear equation, 5x - 5 = 45, since this value makes the equation true as shown. 5x - 5 = 45 5(10) - 5 = 45 Replace x with 10. 50 - 5 = 45 Simplify the left side. 45 = 45 True ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Define a linear equation. 2. Use the addition property of equality. 3. Use the multiplication property of equality. 4. Solve linear equations. 5. Identify linear equations with no solution. 6. Identify linear equations with infinitely many solutions. 7. Troubleshoot common errors. Objective 1 ▶ Define a linear equation. 52


hendricks_intermediate_algebra_1e_ch1_3
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