Question

A line goes through the points (9,8) and (-3,4).

What is the slope of the line? Show your work
Write the equation of the line in point-slope form. Show your work
Write the equation of the line in slope-intercept form. Show your work.

Answer 1

`\textsf{Slope}: \quad (1)/(3)`

`\textsf{Point-slope form}: \quad y-8=(1)/(3)(x-9)`

`\textsf{Slope-intercept form}: \quad y=(1)/(3)x+5`

Explanation:

`\boxed{\begin{minipage}{4 cm}\underline{Slope formula}\\\\slope ($m$) $=(y_2-y_1)/(x_2-x_1)$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ \\are two points on the line.\end{minipage}}`

Define the given points:

• (x₁, y₁) = (9, 8)
• (x₂, y₂) = (-3, 4)

Substitute the defined points into the slope formula:

`\implies \textsf{slope}\:(m)=(y_2-y_1)/(x_2-x_1)=(4-8)/(-3-9)=(-4)/(-12)=(1)/(3)`

`\boxed{\begin{minipage}{4.6 cm}\underline{Point-slope formula}\\\\$y-y_1=m(x-x_1)$\\\\where $m$ is the slope and\\ $(x_1,y_1)$ is a point on the line.\end{minipage}}`

Substitute point (9, 8) and the found slope into the point-slope formula:

`\implies y-8=(1)/(3)(x-9)`

`\boxed{\begin{minipage}{3.8 cm}\underline{Slope-intercept formula}\\\\$y=mx+b$\\\\where $m$ is the slope\\ and $b$ is the $y$-intercept.\end{minipage}}`

To write the equation of the line in slope-intercept form, rearrange the point-slope formula:

`\implies y-8=(1)/(3)(x-9)`

`\implies y-8=(1)/(3)x-3`

`\implies y-8+8=(1)/(3)x-3+8`

`\implies y=(1)/(3)x+5`