Question

graph the equation of a line with a slope of $-\frac{6}{4}$ passes through the coordinate (4, 0):

Explanation:

Step1: Recall Point - Slope Form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $m$ is the slope and $(x_1,y_1)$ is a point on the line. Here, $m =-\frac{6}{4}=-\frac{3}{2}$ and $(x_1,y_1)=(4,0)$.
Substitute these values into the point - slope formula: $y - 0=-\frac{3}{2}(x - 4)$

Step2: Simplify the Equation

Simplify the right - hand side: $y=-\frac{3}{2}x+\frac{3}{2}\times4$.
Calculate $\frac{3}{2}\times4 = 6$, so the equation of the line is $y =-\frac{3}{2}x + 6$.

Step3: Identify Two Points on the Line

We already know that the line passes through $(4,0)$. To find another point, we can use the slope. The slope $m =-\frac{3}{2}$ means that for a run (change in $x$) of $2$, the rise (change in $y$) is $- 3$.
Starting from the point $(4,0)$, if we add $2$ to $x$ (run = 2), we get $x=4 + 2=6$. Then we subtract $3$ from $y$ (rise=-3), so $y=0-3=-3$. So another point on the line is $(6,-3)$.
We can also find the $y$ - intercept by setting $x = 0$ in the equation $y=-\frac{3}{2}x + 6$. When $x = 0$, $y=6$. So the $y$ - intercept is $(0,6)$.

Step4: Graph the Line

1. Plot the points $(4,0)$ and $(0,6)$ (or $(6,-3)$) on the coordinate plane.
2. Draw a straight line passing through these two points. The line should have a negative slope (going down from left to right) since the slope is negative, and it should pass through the point $(4,0)$ and have a $y$ - intercept at $(0,6)$.

Answer:

The equation of the line is $y =-\frac{3}{2}x+6$. To graph it, plot the points $(0,6)$ (y - intercept) and $(4,0)$ (given point), then draw a straight line through them.