The graph of a line in the $xy$ -plane has a slope of $1$ and contains the point $(3,0)$ . The graph of a second line has a slope of $-\frac{1}{4}$ and contains the point $(-7,0)$ . 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

1

and contains the point

(3,0)

. The graph of a second line has a slope of

-\frac{1}{4}

and contains the point

(-7,0)

. If the two lines intersect at the point

(a,b)

, what is the value of

a+b

Write Line Equation: Write the equation of the first line using the slope-point form. $\newline$ The slope-point form of a line is $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is a point on the line. $\newline$ For the first line with a slope of $1$ and passing through the point $(3,0)$ , the equation is: $\newline$ $y - 0 = 1(x - 3)$
Simplify First Line: Simplify the equation of the first line. $\newline$ Simplifying the equation from Step $1$ , we get: $\newline$ $y = x - 3$
Write Second Line: Write the equation of the second line using the slope-point form. $\newline$ For the second line with a slope of $-\frac{1}{4}$ and passing through the point $(-7,0)$ , the equation is: $\newline$ $y - 0 = \left(-\frac{1}{4}\right)(x - (-7))$
Simplify Second Line: Simplify the equation of the second line. $\newline$ Simplifying the equation from Step $3$ , we get: $\newline$ $y = \frac{-1}{4}x + \frac{7}{4}$
Set Equations Equal: Set the equations of the two lines equal to each other to find the $x$ -coordinate of the intersection point. $\newline$ Since the lines intersect at point $(a,b)$ , their $y$ -values are equal at this point. Therefore, we can set the equations equal to each other: $\newline$ $x - 3 = (-\frac{1}{4})x + \frac{7}{4}$
Solve for x: Solve for x. $\newline$ To find the value of $x$ , we combine like terms and solve the equation: $\newline$ $x + \frac{1}{4}x = 3 + \frac{7}{4}$ $\newline$ $\frac{5}{4}x = \frac{19}{4}$ $\newline$ $x = \frac{19}{4} / \frac{5}{4}$ $\newline$ $x = \frac{19}{5}$
Find y-coordinate: Find the y-coordinate of the intersection point by substituting $x$ back into one of the line equations. $\newline$ We can use the first line's equation for simplicity: $\newline$ $y = x - 3$ $\newline$ $y = (19/5) - 3$ $\newline$ $y = (19/5) - (15/5)$ $\newline$ $y = 4/5$
Add Coordinates: Add the $x$ and $y$ coordinates of the intersection point to find $a + b$ . $\newline$ $a = \frac{19}{5}$ $\newline$ $b = \frac{4}{5}$ $\newline$ $a + b = \left(\frac{19}{5}\right) + \left(\frac{4}{5}\right)$ $\newline$ $a + b = \frac{23}{5}$