109 1.1 Linear Equations Identifying Types of Linear Equations 1. If solving a linear equation leads to a true statement such as 0 = 0, the equation is an identity. Its solution set is {all real numbers}. (See Example 3(a).) 2. If solving a linear equation leads to a single solution such as x = 3, the equation is conditional. Its solution set consists of a single element. (See Example 3(b).) 3. If solving a linear equation leads to a false statement such as -3 = 7, the equation is a contradiction. Its solution set is ∅. (See Example 3(c).) Solving for a Specified Variable (Literal Equations) A formula is an example of a literal equation (an equation involving letters). EXAMPLE 4 Solving for a Specified Variable Solve each formula or equation for the specified variable. (a) I = Prt, for t (b) A - P = Prt, for P (c) 312x - 5a2 + 4b = 4x - 2, for x SOLUTION (a) This is the formula for simple interest I on a principal amount of P dollars at an annual interest rate r for t years. To solve for t, we treat t as if it were the only variable, and we treat the other variables as if they were constants. I = Prt I Pr = Prt Pr Divide each side by Pr. I Pr = t, or t = I Pr (b) The formula A - P = Prt is a form of the more common formula A = P11 + rt2, which gives the future value, or maturity value, A of P dollars invested for t years at annual simple interest rate r. A - P = Prt A = P + Prt Transform so that all terms involving P are on one side. A = P11 + rt2 Factor out P. A 1 + rt = P, or P = A 1 + rt Divide by 1 + rt. (c) 312x - 5a2 + 4b = 4x - 2 Solve for x. 6x - 15a + 4b = 4x - 2 Distributive property 6x - 4x = 15a - 4b - 2 Isolate the x-terms on one side. 2x = 15a - 4b - 2 Combine like terms. x = 15a - 4b - 2 2 Divide each side by 2. S Now Try Exercises 39, 47, and 49. Goal: Isolate t on one side. Goal: Isolate P, the specified variable. Pay close attention to this step.
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