Chapter 4.2, Problem 2ACYP

To determine

To Write:

The equation that passes through each of the points :

(-1,12), (4,-8)

Answer to Problem 2ACYP

The required equation of line passing through points (-1,12) , (4,-8) is:

  $y=-4x+8$

Explanation of Solution

Given Information:

The line passes through points (-1,12) , (4,-8)

Formula Used:

The equation of line in slope-intercept form passing through points $(x_1,y_1),(x_2,y_2)$ is given by:

  $\begin{array}{l}y=mx+b\mathrm{...........................}(1)\\ where\\ b=y-\mathrm{int}ercept\\ m=slope=\frac{y_2-y_1}{x_2-x_1}\end{array}$

Calculation:

First , we form the equation :

Let

  $\begin{array}{l}(x_1,y_1)=(-1,12)\\ (x_2,y_2)=(4,-8)\end{array}$

First we find the slope of line containing the given points:

  $\begin{array}{l}m=slope=\frac{y_2-y_1}{x_2-x_1}\\ \Rightarrow m=\frac{-8-12}{4-(-1)}\\ \Rightarrow m=\frac{-20}{5}\\ \Rightarrow m=-4\end{array}$

Substituting the value in equation (1) , we get:

  $\begin{array}{l}y=mx+b\mathrm{.........................}(1)\\ m=-4\\ \Rightarrow y=-4x+b\mathrm{....................}(2)\end{array}$

Using the point $(x,y)=(-1,12)$ to find the y-intercept:

  $\begin{array}{l}y=-4x+b\mathrm{............................}(2)\\ (x,y)=(-1,12)\\ \Rightarrow 12=-4(-1)+b\mathrm{.................}[substituting]\\ \Rightarrow 12=4+b\mathrm{..........................}[simplify]\\ \Rightarrow 12-4=4+b-\mathrm{4...............}[subtract"4"each\_side]\\ \Rightarrow 8=b\\ or\\ b=8\end{array}$

Substituting the value in equation (2) :

  $\begin{array}{l}y=-4x+b\mathrm{.........................}(2)\\ b=8\\ \Rightarrow y=-4x+\mathrm{8.....................}(3)\end{array}$

Hence, the required equation of line passing through points (-1,12) , (4,-8) is:

  $y=-4x+8$