Q50. - Algebra 2

Question

Graph the line that satisfies each set of conditions.

50. perpendicular to the graph of $\mathbf{2}\mathbf{x}+\mathbf{5}\mathbf{y}=\mathbf{10}$, intersects that graph at its y-intercept

Step-by-Step Solution

Answer

The graph of the required line is given as follows:

Step 1 – State the concepts

The y-intercept is the value of y when $x=0$.

 

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 slope of a line perpendicular to a line having slope m is $-\frac{1}{m}$.

 

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

The required line is perpendicular to $2x+5y=10$ and intersects its graph at its y-intercept.

Step 3 – Find the slope

The required line is perpendicular to the line $2x+5y=10$.

 

Converting this straight-line equation to slope-intercept form,

 

$2x+5y=10$  (Given equation)

 

$2x+5y-2x=10-2x$  (Subtract $2x$ from both sides)

 

$5y=-2x+10$  (Simplify and rearrange)

 

$\frac{5y}{5}=\frac{-2x+10}{5}$  (Divide both sides by 5)

 

$y=-\frac{2}{5}x+2$  (Simplify)

 

Comparing with $y=mx+c$, $m=-\frac{2}{5}$ and $c=2$.

 

So, slope of the line $2x+5y=10$ is $m=-\frac{2}{5}$. Then, slope of the required line is $\frac{5}{2}$.

Step 4 – Find the point

Put $x=0$ in $2x+5y=10$ to get,

 

$2\left(0\right)+5y=10$

 

$5y=10$   (Simplify)

 

$\frac{5y}{5}=\frac{10}{5}$  (Divide both sides by 5)

 

$y=2$   (Simplify)

 

So, the y-intercept is 2 and the point through which the required line passes is $\left(0,2\right)$.

Step 5 – Find the equation

Put $m=\frac{5}{2}$ and $\left(h,k\right)=\left(0,2\right)$ in $y-k=m\left(x-h\right)$ to get,

 

$y-2=\frac{5}{2}\left(x-0\right)$

 

$y-2=\frac{5}{2}x$  (Simplify)

 

$2\left(y-2\right)=\frac{5}{2}x\left(2\right)$  (Multiply both sides by 2)

 

$2y-4=5x$   (Simplify)

 

$2y-4-2y=5x-2y$  (Subtract $2y$ from both sides)

 

$5x-2y=-4$ (Simplify and rearrange)

 

So, $5x-2y=-4$ is the equation of the required line.

Step 6 – Graph the equation

Graphing the obtained equation of the straight line,