Question

Write the equation for the line through (-1/2, -7/2) and (2,14) in slope intercept form

Answer 1

SOLUTION

The slope-intercept form for the equation of a line is given by

$\begin{gathered} y=mx+c \\ \text{where m=slope and c=intercept} \end{gathered}$

Giving the point

$\begin{gathered} (-\frac{1}{2},-\frac{7}{2}) \\ \text{and} \\ (2,14) \end{gathered}$

Then

$\begin{gathered} x_1=-\frac{1}{2},y_1=-\frac{7}{2} \\ \text{and } \\ x_2=2,y_2=14 \end{gathered}$

We apply the two point-form for the equation of a line

$\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}$

Substituting the values into the formula, we obtain

$\frac{y-(-\frac{7}{2})}{x-\frac{1}{2}}=\frac{14-(-\frac{7}{2})}{2-(-\frac{1}{2})}$

Simplify the equation above

$\begin{gathered} \frac{y+\frac{7}{2}}{x-\frac{1}{2}}=\frac{14+\frac{7}{2}}{2+\frac{1}{2}} \\ \\ \frac{y+\frac{7}{2}}{x-\frac{1}{2}}=\frac{28+7}{2}\frac{.}{\text{.}}\frac{4+1}{2} \end{gathered}$

Then, change the division to multiplication and take the reciprocal of the last fraction

$\begin{gathered} \frac{y+\frac{7}{2}}{x-\frac{1}{2}}=\frac{35}{2}\times\frac{2}{5} \\ \frac{y+\frac{7}{2}}{x-\frac{1}{2}}=7 \end{gathered}$

Multiply both parts of the equation by the denominator (x-1/2), we obtain

$\begin{gathered} y+\frac{7}{2}=7(x-\frac{1}{2}) \\ \text{expand the parenthesis} \\ y+\frac{7}{2}=7x-\frac{7}{2} \end{gathered}$

Then, make y the subject of the formula

$\begin{gathered} y+\frac{7}{2}=7x-\frac{7}{2} \\ \text{subtract 7/2 from both sides } \\ y=7x-\frac{7}{2}-\frac{7}{2} \\ y=7x-\frac{14}{2} \end{gathered}$

Hence the equation of the line becomes

$\begin{gathered} y=7x-7\ldots..\text{ in slope-intercept } \\ \text{slope}=7\text{ and intercept =-7} \end{gathered}$

Therefore the equation of the line in slope-intercept is

y=7x-7