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Algebra Difficulty: Medium

For the linear function $h$ , the graph of $y = h (x)$ in the xy-plane passes through the points $left parenthesis 7 comma 21 right parenthesis$ and $left parenthesis 9 comma 25 right parenthesis$ . Which equation defines $h$ ?

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Explanation

Choice B is correct. It’s given that the graph of the linear function $h$ , where $y = h (x)$ , passes through the points $left parenthesis 7 comma 21 right parenthesis$ and $left parenthesis 9 comma 25 right parenthesis$ in the xy-plane. An equation defining $h$ can be written in the form $y = m x + b$ , where $y = h (x)$ , $m$ represents the slope of the graph in the xy-plane, and $b$ represents the y-coordinate of the y-intercept of the graph. The slope can be found using any two points, $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , and the formula $m = \frac{(y_{2} - y_{1})}{(x_{2} - x_{1})}$ . Substituting $left parenthesis 7 comma 21 right parenthesis$ and $left parenthesis 9 comma 25 right parenthesis$ for $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , respectively, in the slope formula yields $m = \frac{25 - 21}{9 - 7}$ , which is equivalent to $m = \frac{4}{2}$ , or $m equals 2$ . Substituting $2$ for $m$ and $left parenthesis 7 comma 21 right parenthesis$ for $(x, y)$ in the equation $y = m x + b$ yields $21 = (2) (7) + b$ , or $21 = 14 + b$ . Subtracting $14$ from each side of this equation yields $7 equals b$ . Substituting $2$ for $m$ and $7$ for $b$ in the equation $y = m x + b$ yields $y = 2 x + 7$ . Since $y = h (x)$ , it follows that the equation that defines $h$ is $h (x) = 2 x + 7$ .

Choice A is incorrect. For this function, the graph of $y = h (x)$ in the xy-plane would pass through $left parenthesis 7 comma 0 right parenthesis$ , not $left parenthesis 7 comma 21 right parenthesis$ , and $left parenthesis 9 comma 1 right parenthesis$ , not $left parenthesis 9 comma 25 right parenthesis$ .

Choice C is incorrect. For this function, the graph of $y = h (x)$ in the xy-plane would pass through $left parenthesis 7 comma 70 right parenthesis$ , not $left parenthesis 7 comma 21 right parenthesis$ , and $left parenthesis 9 comma 84 right parenthesis$ , not $left parenthesis 9 comma 25 right parenthesis$ .

Choice D is incorrect. For this function, the graph of $y = h (x)$ in the xy-plane would pass through $left parenthesis 7 comma 88 right parenthesis$ , not $left parenthesis 7 comma 21 right parenthesis$ , and $left parenthesis 9 comma 106 right parenthesis$ , not $left parenthesis 9 comma 25 right parenthesis$ .