Chapter 8.4, Problem 14E

To determine

To find:

The coordinates of two points on the line with the given equation.Then use the points to find the slope of the line.

Answer to Problem 14E

The coordinates of two points on the line would be $(0,3)\text{and (6,0)}$ .

The slope of the linepassing through the given points is $-\frac{1}{2}$ .

Explanation of Solution

Given:

An equation: $x+2y=6$

Concept used:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1},\text{where,}\\ m=\text{Slope of line,}\\ y_2-y_1=\text{Difference between two }y\text{-coordinates}\\ x_2-x_1=\text{Difference between two }x\text{-coordinates of same }y-\text{coordinates}\end{array}$

Calculation:

To find the coordinates of two points on our given line, we will substitute $x=0$ and $y=0$ in our given equation as:

  $\begin{array}{l}x+2y=6\left(\text{Given equation}\right)\\ 0+2y=6\left(\text{Substituting }x=0\right)\\ 2y=6\\ \frac{2y}{2}=\frac{6}{2}\left(\text{Dividing both sides by 2}\right)\\ y=3\end{array}$

  $\begin{array}{l}x+2y=6\left(\text{Given equation}\right)\\ x+2(0)=6\left(\text{Substituting }y=0\right)\\ x+0=6\\ x=6\end{array}$

Therefore, the coordinates of two points on the line would be $(0,3)\text{and (6,0)}$ .

Now we will find the slope of line.

Let point $(0,3)=(x_1,y_1)\text{and (6,0)}=(x_2,y_2)$ . Upon substituting our given coordinates in slope formula, we will get:

  $\begin{array}{l}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{0-3}{6-0}\\ m=\frac{-3}{6}\\ m=-\frac{1}{2}\end{array}$

Therefore, the slope of the line passing through the given points is $-\frac{1}{2}$ .