What is the equation of the line graphed in the $xy$ -plane that passes through the point $(-4,-5)$ and is parallel to the line whose equation is $3x-4y=-8$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $y=-\frac{4}{3}x+10$ $\newline$ (B) $y=\frac{3}{4}x-2$ $\newline$ (C) $y=\frac{3}{4}x-8$ $\newline$ (D) $y=-\frac{4}{3}x-8$

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Find equations of tangent lines using limits

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Q. What is the equation of the line graphed in the

xy

-plane that passes through the point

(-4,-5)

and is parallel to the line whose equation is

3x-4y=-8

\newline

Choose

1

answer:

\newline

(A)

y=-\frac{4}{3}x+10

\newline

(B)

y=\frac{3}{4}x-2

\newline

(C)

y=\frac{3}{4}x-8

\newline

(D)

y=-\frac{4}{3}x-8

Find Slope of Given Line: First, we need to find the slope of the line that is parallel to the given line. Since parallel lines have the same slope, we can find the slope of the given line by rewriting its equation in slope-intercept form $y = mx + b$ , where $m$ is the slope. $\newline$ The equation of the given line is $3x - 4y = -8$ . $\newline$ To find the slope, we need to solve for $y$ :
Isolate y in Equation: Add $4y$ to both sides of the equation to isolate terms with $y$ on one side: $\newline$ $3x - 4y + 4y = -8 + 4y$ $\newline$ This simplifies to: $\newline$ $3x = 4y - 8$
Use Point-Slope Form: Now, divide both sides by $4$ to solve for $y$ : $\newline$ $\frac{3x}{4} = \frac{4y - 8}{4}$ $\newline$ This simplifies to: $\newline$ $y = \frac{3}{4}x - 2$ $\newline$ The slope of the given line is $\frac{3}{4}$ .
Substitute Point and Slope: Since the line we are looking for is parallel to the given line, it will have the same slope, which is $\frac{3}{4}$ . Now we use the point-slope form of the equation of a line, which is $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is a point on the line. $\newline$ We have the point $(-4, -5)$ and the slope $\frac{3}{4}$ .
Distribute Slope: Substitute the point and the slope into the point-slope form: $\newline$ $y - (-5) = \frac{3}{4}(x - (-4))$ $\newline$ This simplifies to: $\newline$ $y + 5 = \frac{3}{4}(x + 4)$
Solve for y: Now distribute the slope $\frac{3}{4}$ through the parentheses: $\newline$ $y + 5 = \left(\frac{3}{4}\right)x + \left(\frac{3}{4}\right)*4$ $\newline$ This simplifies to: $\newline$ $y + 5 = \left(\frac{3}{4}\right)x + 3$
Final Equation: Subtract $5$ from both sides to solve for $y$ : $\newline$ $y = \left(\frac{3}{4}\right)x + 3 - 5$ $\newline$ This simplifies to: $\newline$ $y = \left(\frac{3}{4}\right)x - 2$
Final Equation: Subtract $5$ from both sides to solve for $y$ : $\newline$ $y = \left(\frac{3}{4}\right)x + 3 - 5$ $\newline$ This simplifies to: $\newline$ $y = \left(\frac{3}{4}\right)x - 2$ We have found the equation of the line in slope-intercept form. The final equation is $y = \left(\frac{3}{4}\right)x - 2$ , which corresponds to answer choice $(B)$ .