Question

Write an equation in slope-intercept form of the line that passes through $\left(-4,4\right)$ and $\left(6,9\right)$.

Short answer

The equation in slope-intercept form is $y=\frac{1}{2}x+6$.

Step-by-step solution

Step-1 – Apply the concept of slope-intercept form

Equation of line in slope intercept form is expressed below.

$y=mx+c$

Where m is the slope and c is the intercept of y-axis.

Step-2 – Apply the concept of slope

The slope m of a line passing through two points $\left(x_1,y_1\right)$and $\left(x_2,y_2\right)$is expressed below.

$m=\frac{y_2-y_1}{x_2-x_1}$

Now, compute the slope of the line passing through the points $\left(-4,4\right)$and $\left(6,9\right)$.

$\begin{array}{c}m=\frac{9-4}{6-\left(-4\right)}\\ =\frac{5}{6+4}\\ =\frac{5}{10}\\ =\frac{1}{2}\end{array}$

Therefore, slope of the line passing through the points $\left(-4,4\right)$and $\left(6,9\right)$is $m=\frac{1}{2}$.

Step-3 –Express the equation in point-slope form

The equation in point-slope form is expressed as $y-y_1=m\left(x-x_1\right)$.

Where m is the slope and $\left(x_1,y_1\right)$is the point through which the line passes.

Now, compute the equation of line with slope $m=\frac{1}{2}$and passing through the point $\left(-4,4\right)$.

$\begin{array}{c}y-4=\frac{1}{2}\left(x-\left(-4\right)\right)\\ y-4=\frac{1}{2}\left(x+4\right)\\ y-4=\frac{1}{2}x+2\\ y=\frac{1}{2}x+6\end{array}$

Recall that equation of line in slope intercept form is expressed as $y=mx+c$

Now, the equation is in the form $y=mx+c$. Here slope m of the line is $\frac{1}{2}$ and intercept of y-axis c is $6$.

Hence, the equation of line passing through the points $\left(-4,4\right)$ and $\left(6,9\right)$$y=\frac{1}{2}x+6$.