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 point$\left(1,7\right)$and is perpendicular to the line$x-3y+16=0$

Short answer

(a) The equation of line in slope intercept form is $y=-3x+10$.

(b) The equation of line in general form is $3x+y-10=0$.

(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 point $\left(1,7\right)$.

The line is perpendicular to the line $x-3y+16=0$.

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

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

$y-y_1=m\left(x-x_1\right)\mathrm{...}\left(1\right)$

Here, m is the slope and $x_1,y_1$are the coordinates.

Find the slope of the perpendicular line.

$$\begin{gathered} x-3y+16=0 \\ 3y=x+16 \\ y=\frac{1}{3}x+\frac{16}{3} \end{gathered}$$

So, the slope of the perpendicular line is $\frac{1}{3}$.

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

The relation between the slope of two perpendicular line is given by,

$$\begin{gathered} m_1\cdot m_2=-1 \\ \left(\frac{1}{3}\right)m_2=-1 \\ {m}_2=-3 \end{gathered}$$

Substitute the values in equation (1).

$$\begin{gathered} y-7=-3\left(x-1\right) \\ y=-3x+3+7 \\ y=-3x+10 \end{gathered}$$

So, the equation of line in slope intercept form is $y=-3x+10$.

Part b. Step 1. Write the obtained equation.

The equation in slope intercept form is,

$y=-3x+10\mathrm{...}\left(2\right)$

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.

$$\begin{gathered} y=-3x+10 \\ y+3x-10=0 \\ 3x+y-10=0 \end{gathered}$$

Thus, the equation of line in general form is $3x+y-10=0$.

Part c. Step 1. Write the obtained equation.

The equation in slope intercept form is,

$y=-3x+10\mathrm{...}\left(3\right)$

Part c. Step 2. Find the points.

Substitute 0 for x in equation (3).

$$\begin{gathered} y=-3\left(0\right)+10 \\ =10 \end{gathered}$$

Substitute 0 for y in equation (3).

$$\begin{gathered} 0=-3x+10 \\ 3x=10 \\ x=\frac{10}{3} \end{gathered}$$

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)