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Algebra / Linear equations in two variables Difficulty: Hard

The point with coordinates $left parenthesis d comma 4 right parenthesis$ lies on the line shown. What is the value of $d$ ?

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Explanation

Choice C is correct. It's given from the graph that the points $left parenthesis 0 comma 7 right parenthesis$ and $left parenthesis 8 comma 0 right parenthesis$ lie on the line. For two points on a line, $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , the slope of the line can be calculated using the slope formula $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . Substituting $left parenthesis 0 comma 7 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis 8 comma 0 right parenthesis$ for $left parenthesis x 2 comma y 2 right parenthesis$ in this formula, the slope of the line can be calculated as $m = \frac{0 - 7}{8 - 0}$ , or $m = - \frac{7}{8}$ . It's also given that the point $left parenthesis d comma 4 right parenthesis$ lies on the line. Substituting $left parenthesis d comma 4 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ , $left parenthesis 8 comma 0 right parenthesis$ for $left parenthesis x 2 comma y 2 right parenthesis$ , and $- \frac{7}{8}$ for $m$ in the slope formula yields $- \frac{7}{8} = \frac{0 - 4}{8 - d}$ , or $- \frac{7}{8} = \frac{- 4}{8 - d}$ . Multiplying both sides of this equation by $8 minus d$ yields $- \frac{7}{8} (8 - d) = - 4$ . Expanding the left-hand side of this equation yields $- 7 + \frac{7}{8} d = - 4$ . Adding $7$ to both sides of this equation yields $seven eighths d equals 3$ . Multiplying both sides of this equation by $\frac{8}{7}$ yields $d = \frac{24}{7}$ . Thus, the value of $d$ is $StartFraction 24 Over 7 EndFraction$ .

Choice A is incorrect. This is the value of $y$ when $x equals 4$ .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.