The graph of a line in the $xy$ -plane has a slope of $4$ and contains the point $(1,-5)$ . The graph of a second line passes through the points $(0,4)$ and $(12,0)$ . If the two lines intersect at the point $(a,b)$ , what is the value of $a-b$ ?

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Q. The graph of a line in the

xy

-plane has a slope of

4

and contains the point

(1,-5)

. The graph of a second line passes through the points

(0,4)

and

(12,0)

. If the two lines intersect at the point

(a,b)

, what is the value of

a-b

Find First Line Equation: Find the equation of the first line with a slope of $4$ that passes through the point $(1, -5)$ . $\newline$ Using the slope-intercept form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept, we can substitute the given point to find $b$ . $\newline$ $y = mx + b$ $\newline$ $-5 = 4(1) + b$ $\newline$ $-5 = 4 + b$ $\newline$ $b = -5 - 4$ $\newline$ $(1, -5)$ $0$ $\newline$ So, the equation of the first line is $(1, -5)$ $1$ .
Find Second Line Slope: Find the slope of the second line that passes through the points $(0, 4)$ and $(12, 0)$ . The slope $m$ is given by the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ . $m = \frac{0 - 4}{12 - 0}$ $m = \frac{-4}{12}$ $m = -\frac{1}{3}$ So, the slope of the second line is $-\frac{1}{3}$ .
Find Second Line Equation: Find the equation of the second line using the slope and one of the points, say $(0, 4)$ . $\newline$ Since the line passes through $(0, 4)$ , the y-intercept $b$ is $4$ . $\newline$ The equation of the second line is $y = (-1/3)x + 4$ .
Find Intersection Point X-coordinate: Set the equations of the two lines equal to each other to find the x-coordinate $a$ of the intersection point. $4x - 9 = \left(-\frac{1}{3}\right)x + 4$ To solve for $x$ , we combine like terms. $4x + \left(\frac{1}{3}\right)x = 4 + 9$ $\left(\frac{12}{3}\right)x + \left(\frac{1}{3}\right)x = 13$ $\left(\frac{13}{3}\right)x = 13$ $x = \frac{13}{\left(\frac{13}{3}\right)}$ $x = 13 \times \left(\frac{3}{13}\right)$ $x = 3$ So, the x-coordinate of the intersection point is $3$ .
Find Intersection Point Y-coordinate: Substitute $x = 3$ into one of the line equations to find the y-coordinate ( $b$ ) of the intersection point. We can use the first line's equation for this. $\newline$ $y = 4x - 9$ $\newline$ $y = 4(3) - 9$ $\newline$ $y = 12 - 9$ $\newline$ $y = 3$ $\newline$ So, the y-coordinate of the intersection point is $3$ .
Calculate $a - b$ : Calculate $a - b$ using the coordinates of the intersection point $(a, b)$ which are $(3, 3)$ . $\newline$ $a - b = 3 - 3$ $\newline$ $a - b = 0$ $\newline$ So, the value of $a - b$ is $0$ .