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Algebra / Linear equations in two variables Difficulty: Hard

$a, x, plus b y, equals b$

In the equation above, a and b are constants and $0 is less than a, which is less than b$ . Which of the following could represent the graph of the equation in the xy-plane?

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Explanation

Choice C is correct. The given equation $a, x plus b y, equals b$ can be rewritten in slope-intercept form, $y equals, m x plus k$ , where m represents the slope of the line represented by the equation, and k represents the y-coordinate of the y-intercept of the line. Subtracting ax from both sides of the equation yields $b y equals, negative a, x plus b$ , and dividing both sides of this equation by b yields $y equals, the negative of the fraction a over b, end fraction, times x, plus, the fraction b over b, end fraction$ , or $y equals, the negative of the fraction a over b, end fraction, times x, plus 1$ . With the equation now in slope-intercept form, it shows that $k equals 1$ , which means the y-coordinate of the y-intercept is 1. It’s given that a and b are both greater than 0 (positive) and that $a is less than b$ . Since $m equals, the negative of the fraction a over b$ , the slope of the line must be a value between $negative 1$ and 0. Choice C is the only graph of a line that has a y-value of the y-intercept that is 1 and a slope that is between $negative 1$ and 0.

Choices A, B, and D are incorrect because the slopes of the lines in these graphs aren’t between $negative 1$ and 0.