Q22. - Algebra 1

Question

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

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

Step-by-Step Solution

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 1. Write the equation of the line that passes through the point x 1 , y 1 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 − 1 3 .

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 $y=\frac{-1}{3}x-\frac{1}{3}$.