Chapter 1, Problem 47RE

To determine

To graph:the line with the given slope and the point.

Answer to Problem 47RE

The equation of the line is $y=\frac{2}{3}x+\frac{4}{3}$ .

  

Explanation of Solution

Given:

Slope $\frac{2}{3}$ containing the point $\left(1,2\right)$

Formula Used:

The equation of a non-verticalline : $y-y_1=m\left(x-x_1\right)$

Calculation:

An equation of a nonvertical line with slope $m$ that contains the points $\left(x_1,y_1\right)$ is $y-y_1=m\left(x-x_1\right)$

So, the equation of the line can be determined using the point $\left(1,2\right)$ and $m=\frac{2}{3}$ .

  $y-2=\left(\frac{2}{3}\right)\left(x-1\right)$

  $y-2=\frac{2}{3}x-\frac{2}{3}$

Multiply by 3 on both sides of the equation to clear the fractions.

  $3\left(y-2\right)=3\cdot \frac{2}{3}x-3\cdot \frac{2}{3}$

  $3y-6=2x-2$

Subtract 6 from both sides of the equation.

  $3y-6+6=2x-2+6$

  $3y=2x+4$

Divide by 3 on both sides of the equation.

  $\frac{3y}{3}=\frac{2x}{3}+\frac{4}{3}$

  $y=\frac{2}{3}x+\frac{4}{3}$

Therefore, the equation of the line is $y=\frac{2}{3}x+\frac{4}{3}$ .

Substitute 1 for x in the equation to get a point on the line.

  $y=\frac{2}{3}\cdot 1+\frac{4}{3}$

  $=\frac{6}{2}$

  $=2$

For getting the next point on the line, start from the point $\left(1,2\right)$ , move 3 units to the right and then up 2 units. We get the new point $\left(4,4\right)$ .

The line can be graphed by plotting the points and drawing a line passing through them.

  

Conclusion:

Therefore, the equation of the line is $y=\frac{2}{3}x+\frac{4}{3}$ .