Question

Write an equation of the line that passes through the given point and has the given slope.

$\left(5,-2\right)$, slope$-\frac{1}{3}$

Short answer

The equation of the line that passes through the point $\left(5,-2\right)$and has the slope $-\frac{1}{3}$ is $y=\frac{-1}{3}x-\frac{1}{3}$.

Step-by-step solution

Step 1. Write the equation of the line that passes through the point x1,y1 and has the slope m.

The equation of the line that passes through the point$\left(x_1,y_1\right)$ and has the slope$m$ is given by:

$y-y_1=m\left(x-x_1\right)$

Step 2. Write the equation of the line that passes through the point 5,−2 and has the slope −13.

The slope of the line is $-\frac{1}{3}$. Therefore, the value of$m$ is $-\frac{1}{3}$.

The given point is $\left(5,-2\right)$. Therefore, the values of$x_1$and$y_1$are 5 and$-2$respectively.

Now substitute$-\frac{1}{3}$for $m$, 5 for$x_1$and$-2$for$y_1$in the equation$y-y_1=m\left(x-x_1\right)$to find the equation of the line.

$$\begin{gathered} y-y_1=m\left(x-x_1\right) \\ y-\left(-2\right)=-\frac{1}{3}\left(x-5\right) \\ y+2=\frac{-1}{3}x+\frac{5}{3} \\ y=\frac{-1}{3}x+\frac{5}{3}-2 \\ y=\frac{-1}{3}x+\frac{5-6}{3} \\ y=\frac{-1}{3}x-\frac{1}{3} \end{gathered}$$

Therefore, the equation of the line that passes through the point$\left(5,-2\right)$and has the slope$-\frac{1}{3}$is $y=\frac{-1}{3}x-\frac{1}{3}$.