Chapter 4.3, Problem 50HP

To determine

To find: The equation which is correct from these two equations $y-7=-\frac{11}{9}\left(x+3\right)$ and $y-4=-\frac{9}{11}\left(x+6\right)$ if the line passes through the points $\left(3,-7\right)$ and $\left(-6,4\right)$ . Explain if each of these two equations are correct.

Answer to Problem 50HP

No, the both equations $y-7=-\frac{11}{9}\left(x+3\right)$ and $y-4=-\frac{9}{11}\left(x+6\right)$ are wrong. The correct equation is $y+7=-\frac{11}{9}\left(x-3\right)$ .

Explanation of Solution

Given information: The given equations are,

  $y-7=-\frac{11}{9}\left(x+3\right)\mathrm{......}\left(1\right)$

  $y-4=-\frac{9}{11}\left(x+6\right)\mathrm{......}\left(2\right)$

Also, the line passes through the points $\left(3,-7\right)$ and $\left(-6,4\right)$ .

Calculation: Using the formula, the slope $\left(m\right)$ of line through the points $\left(3,-7\right)$ and $\left(-6,4\right)$ is evaluated as,

  $\begin{array}{c}m=\frac{4-\left(-7\right)}{-6-3}\mathrm{......}\left(\begin{array}{l}\text{Using formula,}\\ \text{Slope}\left(m\right)=\frac{y_2-y_1}{x_2-x_1}\end{array}\right)\\ =-\frac{11}{9}\end{array}$

Now, the equation of the line through the points $\left(3,-7\right)$ using slope-point form is evaluated as,

  $\begin{array}{c}y-\left(-7\right)=-\frac{11}{9}\left(x-3\right)\mathrm{......}\left(\begin{array}{l}\text{Slope-point form}\\ y-y_1=m\left(x-x_1\right)\end{array}\right)\\ y+7=-\frac{11}{9}\left(x-3\right)\mathrm{......}\left(3\right)\end{array}$

Now, the line (3) doesn’t match with (1) and (2). Therefore, both the equations (1) and (2) are wrong.

Hence the line (3), $y-7=-\frac{11}{9}\left(x+3\right)$ is correct.