Question

Write an equation in slope-intercept form for the line that passes through the given point and is parallel to the graph of the given equation.

$\left(0,3\right)\mathbf{,}\mathit{y}\mathbf{=}\mathbf{3}\mathit{x}\mathbf{+}\mathbf{5}$

Short answer

The required equation of the line in slope-intercept form is$\mathit{y}\mathbf{=}\mathbf{3}\mathit{x}\mathbf{+}\mathbf{3}$

Step-by-step solution

Step 1. Slope-intercept form of an equation of a line:

The slope-intercept form of an equation of a line is given by

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

Where , slope $=m$, and, $y-$axis intercept$=b$.

Step 2. Write an equation of a line passing through a point (x1,y1) and having slope m.

The equation of a line passing through a point$\left(x_1,y_1\right)$ and having slope$m$ is given byrole="math" localid="1647430983632" $y-y_1=m\left(x-x_1\right)\mathrm{..........}\left(2\right)$

Step 3. Write the equation of the line that passes through the point (2,5) and is parallel to the line y=x−3. 

The equation of the given line is,

$y=3x+5........\left(3\right)$

Compare (3) with (1).

The slope of the line $\left(3\right)=3$

Since, the unknown line is parallel to the line (3).

Since, two lines are parallel to each other if the slope of one of the line is equal to the slope of the other line.

So, the slope of the unknown line $\left(m\right)=$the slope of the line (3) $=3$

Since, the unknown line passes through the point $\left(0,3\right)$

Therefore, substitute $x_1=0,y_1=3$and $m=3$, in (2)

$\begin{array}{c}y-3=3\left(x-0\right)\\ y-3=3x\\ y=3x+3\end{array}$

This is the equation of the unknown line in slope-intercept form $y=mx+b$, where $m=3$and $b=3$.

Therefore the required equation of the line in slope-intercept form is $y=3x+3$.