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Algebra Difficulty: Hard

$x$ $x$	$y$ $y$
$k$ $k$	$13$ $13$
$k + 7$ $k plus 7$	$- 15$ $negative 15$

The table gives the coordinates of two points on a line in the xy-plane. The y-intercept of the line is $left parenthesis k minus 5 comma b right parenthesis$ , where $k$ and $b$ are constants. What is the value of $b$ ?

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Explanation

The correct answer is $33$ . It’s given in the table that the coordinates of two points on a line in the xy-plane are $left parenthesis k comma 13 right parenthesis$ and $(k + 7, - 15)$ . The y-intercept is another point on the line. The slope computed using any pair of points from the line will be the same. The slope of a line, $m$ , between any two points, $left parenthesis x 1 comma y 1 right parenthesis$ and $left parenthesis x 2 comma y 2 right parenthesis$ , on the line can be calculated using the slope formula, $m = \frac{(y_{2} - y_{1})}{(x_{2} - x_{1})}$ . It follows that the slope of the line with the given points from the table, $left parenthesis k comma 13 right parenthesis$ and $(k + 7, - 15)$ , is $m = \frac{- 15 - 13}{k + 7 - k}$ , which is equivalent to $m = \frac{- 28}{7}$ , or $m = - 4$ . It's given that the y-intercept of the line is $left parenthesis k minus 5 comma b right parenthesis$ . Substituting $negative 4$ for $m$ and the coordinates of the points $left parenthesis k minus 5 comma b right parenthesis$ and $left parenthesis k comma 13 right parenthesis$ into the slope formula yields $- 4 = \frac{13 - b}{k - (k - 5)}$ , which is equivalent to $- 4 = \frac{13 - b}{k - k + 5}$ , or $- 4 = \frac{13 - b}{5}$ . Multiplying both sides of this equation by $5$ yields $- 20 = 13 - b$ . Subtracting $13$ from both sides of this equation yields $- 33 = - b$ . Dividing both sides of this equation by $negative 1$ yields $b equals 33$ . Therefore, the value of $b$ is $33$ .