164 Chapter 2 Linear Equations and Inequalities Section 2.6 Literal Equations and Applications of Geometry Classroom Examples: p. 170, Exercises 16 and 24 1. Formulas and Literal Equations A literal equation is an equation that has more than one variable. A formula is a literal equation with a specific application. For example, the perimeter of a triangle (distance around the triangle) can be found by the formula P a b c, where a, b, and c are the lengths of the sides (Figure 2-7). b 7 ft P a b c To solve a formula for a different variable, we use the same properties of equality outlined in the earlier sections of this chapter. For example, consider the two equations 2x 3 11 and wx y z. Suppose we want to solve for x in each case: 2x 3 11 wx y z 2x 3 3 11 3 wx y y z y Subtract 3. Subtract y. 2x 8 wx z y Divide by 2. Divide by w. The equation on the left has only one variable and we are able to simplify the equation to find a numerical value for x. The equation on the right has multiple variables. Because we do not know the values of w, y, and z, we are not able to simplify further. The value of x is left as a formula in terms of w, y, and z. Solving for an Indicated Variable Example 1 Solve for the indicated variables. a. d rt for t b. 5x 2y 12 for y Solution: a. for t The goal is to isolate the variable t. Because the relationship between r and t is multiplication, we reverse the process by dividing both sides by r. or equivalently t d r d rt d r t, d r rt r x z y w x 4 wx w z y w 2x 2 8 2 Answers 1. 2. a 7 4b 2 or a 4b 7 2 l A w a 5 ft c ? b 7 ft If the perimeter is 20 ft, then c P a b 20 ft 5 ft 7 ft 8 ft Figure 2-8 a 5 ft c 8 ft 5 ft 7 ft 8 ft 20 ft Figure 2-7 In this section, we will learn how to rewrite formulas to solve for a different variable within the formula. Suppose, for example, that the perimeter of a triangle is known and two of the sides are known (say, sides a and b).Then the third side, c, can be found by subtracting the lengths of the known sides from the perimeter (Figure 2-8). Concepts 1. Formulas and Literal Equations 2. Geometry Applications Skill Practice Solve for the indicated variable. 1. A lw for l 2. 2a 4b 7 for a
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