Q35. - Algebra 2

Question

Write an equation in slope-intercept form for the line that satisfies each set of conditions.

35. passes through $\left(\mathbf{4},\mathbf{6}\right)$, parallel to the graph of $\mathbf{y}=\frac{\mathbf{2}}{\mathbf{3}}\mathbf{x}+\mathbf{5}$

Step-by-Step Solution

Answer

The equation of the required straight line in slope-intercept form is $y=\frac{2}{3}x+\frac{10}{3}$.

Step 1 – State the concept

The slope intercept form of a straight-line equation is $y=mx+c$ where $m$ is the slope and $c$ is the y-intercept.

The slopes of parallel lines are equal.

The equation of a straight-line having slope $m$ and passing through the point $\left(h,k\right)$ is given as $y-k=m\left(x-h\right)$.

Step 2 – List the given data

It is given that the line passes through $\left(4,6\right)$ and is parallel to $y=\frac{2}{3}x+5$.

Then, $\left(h,k\right)=\left(4,6\right)$.

Step 3 – Determine the slope

Comparing $y=\frac{2}{3}x+5$ to $y=mx+c$, $m=\frac{2}{3}$.

So, the slope of $y=\frac{2}{3}x+5$ and thus of the required line, as they are parallel, is $\frac{2}{3}$.

Then, $m=\frac{2}{3}$.

Step 4 – Write the equation

Put $m=\frac{2}{3}$ and $\left(h,k\right)=\left(4,6\right)$ in $y-k=m\left(x-h\right)$ to get,

$y-6=\frac{2}{3}\left(x-4\right)$

$y-6=\frac{2}{3}x-\frac{8}{3}$  (Distributive property)

$y-6+6=\frac{2}{3}x-\frac{8}{3}+6$  (Add 6 to both sides)

$y=\frac{2}{3}x+\frac{10}{3}$  (Simplify)

So, the required equation of the straight line in slope-intercept form is $y=\frac{2}{3}x+\frac{10}{3}$.