207 2.1 Rectangular Coordinates and Graphs SOLUTION The year 2014 lies halfway between 2011 and 2017, so we must find the coordinates of the midpoint of the line segment that has endpoints 12011, 167.72 and 12017, 233.72. (Here, the second component is in billions of dollars.) M= a 2011 + 2017 2 , 167.7 + 233.7 2 b = 12014, 200.72 Use the midpoint formula. Our estimate is $200.7 billion, which is greater than the actual figure of $195.4 billion. Models are used to predict outcomes. They rarely give exact values. S Now Try Exercise 41. Equations in Two Variables Ordered pairs are used to express the solutions of equations in two variables. When an ordered pair represents the solution of an equation with the variables x and y, the x-value is written first. For example, we say that 11, 22 is a solution of 2x - y = 0. Substituting 1 for x and 2 for y in the equation gives a true statement. 2x - y = 0 2112 - 2≟0 Let x = 1 and y = 2. 0 = 0 ✓ True EXAMPLE 7 Finding Ordered-Pair Solutions of Equations For each equation, find at least three ordered pairs that are solutions. (a) y = 4x - 1 (b) x = 2y - 1 (c) y = x2 - 4 SOLUTION (a) Choose any real number for x or y, and substitute in the equation to obtain the corresponding value of the other variable. For example, let x = -2 and then let y = 3. y = 4x - 1 y = 41-22 - 1 Let x = -2. y = -8 - 1 Multiply. y = -9 Subtract. y = 4x - 1 3 = 4x - 1 Let y = 3. 4 = 4x Add 1. 1 = x Divide by 4. This gives the ordered pairs 1-2, -92 and 11, 32. Verify that the ordered pair 10, -12 is also a solution. (b) x = 2y - 1 Given equation 1 = 2y - 1 Let x = 1. 1 = y - 1 Square each side. 2 = y Add 1. One ordered pair is 11, 22. Verify that the ordered pairs 10, 12 and 12, 52 are also solutions of the equation.
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