Question

Lines A description of line is give. (a) Find am equation of line in the slope intercept form. (b) Find an equation for the line in general form. (c) Graph the line.

The line that passes through the origin and is parallel to the line containing the point$\left(2,4\right)$ and $\left(4,-4\right)$

Short answer

(a) The equation of line in slope intercept form is $y=-4x$.

(b) The equation of line in general form is $4x+y=0$.

(c) The graph of the line is shown in Figure (1).

Step-by-step solution

Part a. Step 1. Write the given information.

The line passes through the origin.

The line is parallel to the line passing through the points$\left(2,4\right)$ and $\left(4,-4\right)$.

Part a. Step 2. Write standard equation of line in slope intercept form.

The equation of line in slope intercept form is given by,

$y=mx+b\mathrm{...}\left(1\right)$

Here, m is the slope and b is the y intercept.

The slope of the line that passes through two given points is calculated as,

$$\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ =\frac{\left(-4\right)-4}{4-2} \\ =\frac{-8}{2} \\ =-4 \end{gathered}$$

Part a. Step 3. Find the equation of line in slope intercept form.

It is known that two parallel lines have the same slope.

So, the slope of line passing through origin is $m=-4$.

The equation of line passing through the origin is given by,

$y=mx$

Substitute$-4$ for m in above equation.

$y=-4x$

So, the equation of line in slope intercept form is $y=-4x$.

Part b. Step 1. Write the obtained equation.

The equation in slope intercept form is,

$y=-4x\mathrm{...}\left(2\right)$

Part b. Step 2. Write the equation of line in general form.

The equation of line in general form is given by,

$ax+by+c=0$

Here, x, y are the coordinate and a,b,c are the constant.

Part b. Step 3. Find the equation of line general form.

In equation (1) move all the terms to one side of the equation.

$$\begin{gathered} y=-4x \\ 4x+y=0 \end{gathered}$$

Thus, the equation of line in general form is $4x+y=0$.

Part c. Step 1. Write the obtained equation.

The equation in slope intercept form is,

$y=-4x\mathrm{...}\left(3\right)$

Part c. Step 2. Find the points.

Substitute 0 for x in equation (3).

$$\begin{gathered} y=-4\left(0\right) \\ =0 \end{gathered}$$

Substitute $-1$ for x in equation (3).

$$\begin{gathered} y=-4\left(-1\right) \\ =4 \end{gathered}$$

Part c. Step 3. Draw the graph.

Plot the points on the graph and join them as shown in figure.

Figure (1)

Thus, the graph of the line is shown in Figure (1)