Question

graph the line that has a slope of $-\frac{2}{3}$ and includes the point (3, 2). click to select points on the graph.

Explanation:

Step1: Recall the point - slope form of a line

The point - slope form of a linear equation is $y - y_1=m(x - x_1)$, where $m$ is the slope and $(x_1,y_1)$ is a point on the line. We know that $m =-\frac{2}{3}$ and $(x_1,y_1)=(3,2)$. Substituting these values into the point - slope form, we get $y - 2=-\frac{2}{3}(x - 3)$.

Step2: Simplify the equation

First, distribute the $-\frac{2}{3}$ on the right - hand side: $y - 2=-\frac{2}{3}x+2$. Then, add 2 to both sides of the equation to get the slope - intercept form $y=-\frac{2}{3}x + 4$.

Step3: Find another point on the line

We can use the slope to find another point. The slope $m =-\frac{2}{3}=\frac{\text{change in }y}{\text{change in }x}$. Starting from the point $(3,2)$, if we increase $x$ by 3 (the denominator of the slope), $x$ becomes $3 + 3=6$. Then, the change in $y$ is $- 2$ (the numerator of the slope), so $y$ becomes $2+( - 2)=0$. So the point $(6,0)$ is on the line. Also, if we decrease $x$ by 3, $x = 3-3 = 0$, and the change in $y$ is $+2$, so $y=2 + 2=4$, giving the point $(0,4)$.

Step4: Graph the line

Plot the points $(3,2)$, $(0,4)$ and $(6,0)$ (or other points found using the slope) and draw a straight line through them.

Answer:

To graph the line:

1. Plot the point $(3,2)$ (given point).
2. Use the slope $-\frac{2}{3}$: from $(3,2)$, move 3 units to the right (increase $x$ by 3) and 2 units down (decrease $y$ by 2) to get the point $(6,0)$; or move 3 units to the left (decrease $x$ by 3) and 2 units up (increase $y$ by 2) to get the point $(0,4)$.
3. Draw a straight line through the plotted points (e.g., $(0,4)$, $(3,2)$, $(6,0)$).