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Algebra Difficulty: Hard

The figure presents the graph of a line in a coordinate plane, with the origin labeled O. The horizontal axis is labeled “Number of hours at job A,” and the numbers 5 through 20, in increments of 5, are indicated. The vertical axis is labeled “Number of hours at job B,” and the numbers 5 and 10 are indicated. The line begins on the vertical axis at 8, and slants downward and to the right until it ends on the horizontal axis at 16.

To earn money for college, Avery works two part-time jobs: A and B. She earns $10 per hour working at job A and $20 per hour working at job B. In one week, Avery earned a total of s dollars for working at the two part-time jobs. The graph above represents all possible combinations of numbers of hours Avery could have worked at the two jobs to earn s dollars. What is the value of s ?

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Explanation

Choice B is correct. Avery earns $10 per hour working at job A. Therefore, if she works a hours at job A, she will earn $10 a$ dollars. Avery earns $20 per hour working at job B. Therefore, if she works b hours at job B, she will earn $20 b$ dollars. The graph shown represents all possible combinations of the number of hours Avery could have worked at the two jobs to earn s dollars. Therefore, if she worked a hours at job A, worked b hours at job B, and earned s dollars from both jobs, the following equation represents the graph: $10 a, plus 20 b, equals s$ , where s is a constant. Identifying any point $with coordinates a, comma b$ from the graph and substituting the values of the coordinates for a and b, respectively, in this equation yield the value of s. For example, the point $with coordinates 16 comma 0$ , where $a, equals 16$ and $b equals 0$ , lies on the graph. Substituting 16 for a and 0 for b in the equation $10 a, plus 20 b, equals s$ yields $10 times 16, plus, 20 times 0, equals s$ , or $160 equals s$ . Similarly, the point $with coordinates 0 comma 8$ , where $a, equals 0$ and $b equals 8$ , lies on the graph. Substituting 0 for a and 8 for b in the equation $10 a, plus 20 b, equals s$ yields $10 times 0, plus, 20 times 8, equals s$ , or $160 equals s$ .

Choices A, C, and D are incorrect. If the value of s is 128, 200, or 320, then no points $with coordinates a, comma b$ on the graph will satisfy this equation. For example, if the value of s is 128 (choice A), then the equation $10 a, plus 20 b, equals s$ becomes $10 a, plus 20 b, equals 128$ . The point $the point with coordinates 16 comma 0$ , where $a, equals 16$ and $b equals 0$ , lies on the graph. However, substituting 16 for a and 0 for b in $10 a, plus 20 b, equals s$ yields $10 times 16, plus, 20 times 0, equals 128$ , or $160 equals 128$ , which is false. Therefore, $the point with coordinates 16 comma 0$ doesn’t satisfy the equation, and so the value of s can’t be 128. Similarly, if $s equals 200$ (choice C) or $s equals 320$ (choice D), then substituting 16 for a and 0 for b yields $160 equals 200$ and $160 equals 320$ , respectively, which are both false.