298 CHAPTER 6 Algebra, Graphs, and Functions In this section, we discuss solving linear (or first-degree) equations . A linear equation in one variable is one in which the largest exponent on the variable is 1. Examples of linear equations are − = x5 1 3 and + = − x x 2 4 6 5. Equivalent equations are equations that have the same solution. The equations − = = x x 2 5 1, 2 6, and = x 3 are all equivalent equations, since they all have the same solution, 3. When we solve an equation, we write the given equation as a series of simpler equivalent equations until we obtain an equation of the form = x c, where c is some real number. To solve any equation, we have to isolate the variable . That means getting the variable by itself on one side of the equal sign. The four properties of equality that we are about to discuss are used to isolate the variable. The first is the addition property. Example 3 Checking a Solution Determine whether −1 is a solution to the equation + − − = − − + x x x x 7( 2) 5 28 4( 3) 3( 1). Solution To determine whether −1 is a solution to the equation, we substitute −1 for each x in the equation. We then use the order of operations to evaluate the expression on each side of the equation. If the two sides of the equation are equal, then −1 is a solution to the equation. + − − = − − + −+ −−− = −− −−+ − − − = − − + − = − − − = − x x x x 7( 2) 5 28 4( 3) 3( 1) 7(12)5(1)28 4(13)3(11) 7(1) 5( 1) 28 4( 4) 3(0) 7 5 28 16 0 16 16 ? ? ? Substituted 1− for x. Evaluated expressions within parentheses. Performed multiplications. True Because −1 makes the equation a true statement, −1 is a solution to the equation. 7 Now try Exercise 43 Did You Know? An Important Breakthrough This painting by Charles Demuth, called I Saw the Figure 5 in Gold, depicts the abstract nature of numbers. Mathematician and philosopher Bertrand Russell observed in 1919 that it must have required “many ages to discover that a brace of pheasants and a couple of days were both instances of the number 2.” The discovery that numbers could be used not only to count objects such as the number of birds but also to represent abstract quantities represented a breakthrough in the development of algebra. Addition Property of Equality If = a b, then + = + a c b c for all real numbers a b , , and c. The addition property of equality indicates that the same number can be added to both sides of an equation without changing the solution. Example 4 Using the Addition Property of Equality Determine the solution to the equation − = x 8 12. Solution To isolate the variable, add 8 to both sides of the equation. x x x x 8 12 8 8 12 8 0 20 20 − = − + = + + = = CHECK: x 8 12 208 12 12 12 ? − = − = = Substitute 20 for x. True 7 Now try Exercise 45 Alfonso Vicente/Alamy Stock Photo
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