# What equation in slope intercept form represents the line that passes through the two points (-3,...

## Question:

What equation in slope intercept form represents the line that passes through the two points (-3, 1) and (4, 6)?

## Point-Slope Form and Slope-Intercept Form:

(i) The point-slope form of a line of slope {eq}m {/eq} that passes through a point {eq}(x_1,y_1) {/eq} is given by:

$$y-y_1=m(x-x_1)$$

(ii) The slope-intercept form of a line of slope {eq}m {/eq} and y-intercept {eq}b {/eq} is given by:

$$y=mx+b$$

The equation of point-slope form can be converted to the equation of slope-intercept form using algebraic operations.

The line passes through the points:

$$(-3, 1)=(x_1, y_1) \\ (4, 6)= (x_2, y_2)$$

Its slope is found using:

\begin{align} m&= \dfrac{y_2-y_1}{x_2-x_1} \\[0.3cm] &= \dfrac{6-1}{4-(-3)} \\[0.3cm] &= \dfrac{5}{7} \\[0.3cm] \end{align}

The equation of the line is found using:

$$y-y_1=m(x-x_1) \\[0.3cm] y-1= \dfrac{5}{7} (x-(-3)) \\ y-1 = \dfrac{5}{7} (x+3) \\ y-1 = \dfrac{5}{7} x + \dfrac{15}{7} \\ \text{Adding 1 on both sides}, \\ y= \dfrac{5}{7} x + \dfrac{15}{7} +1 \\ y= \dfrac{5}{7} x + \dfrac{15}{7} + \dfrac{7}{7} \\ \boxed{\mathbf{y= \dfrac{5}{7} x + \dfrac{22}{7} }}$$

This is of the form {eq}y=mx+b {/eq} and hence it is in the slope-intercept form.