Lines are perpendicular when their slopes are the negative recicprocals of each other such as \(\displaystyle 2\hspace{1mm}and\hspace{1mm}-\frac{1}{2}\). To find the slope of our equation we must change it to slope y-intercept form.

\(\displaystyle 4y+3x=8\)

Subtract the x variable from both sides:

\(\displaystyle 4y=-3x+8\)

Divide by 4 to isolate y:

\(\displaystyle y=\frac{-3}{4}x+2\)

The negative reciprocal of the above slope:  \(\displaystyle \frac{-3}{4} \rightarrow \frac{4}{3}\). The only equation with this slope is \(\displaystyle y=\frac{4}{3}x-7\). 

To find where this equation crosses the \(\displaystyle y\) axis or its \(\displaystyle y\)-intercept, change the equation into slope intercept form.

\(\displaystyle 6x-3y=9\)

Subtract to isolate \(\displaystyle y\):

\(\displaystyle -3y=-6x+9\)

Divide both sides by \(\displaystyle -3\) to completely isolate \(\displaystyle y\):

\(\displaystyle y=2x-3\)

This form is the slope intercept form \(\displaystyle y=mx+b\) where \(\displaystyle m\) is the slope of the line and \(\displaystyle b\) is the \(\displaystyle y\)-intercept.

\(\displaystyle 3x-4y=-24\)

To find the x-intercepts, remember that the line is crossing the x-axis, and that y=0 when the line crosses the x-axis.

So plug in y=0 into the equation above.

\(\displaystyle 3x-4(0)=-24\)

\(\displaystyle 3x=-24\)

\(\displaystyle x=-8\)

To find the y-intercepts, remember that the line is crossing the y-axis, and that x=0 when the line crosses the x-axis.

So plug in x=0 into the equation above.

\(\displaystyle 3(0)-4y=-24\)

\(\displaystyle -4y=-24\)

\(\displaystyle y=6\)

Slope is the change of a line. To find this line one can remember it as rise over run. This rise over run is really the change in the y direction over the change in the x direction.

Therefore the formula for slope is as follows.

\(\displaystyle Slope = \frac{\Delta Y}{\Delta X}=\frac{y_2-y_1}{x_2-x_1}\)

Plugging in our given points 

\(\displaystyle (x_1,y_1)=(-2,7)\) and \(\displaystyle (x_2,y_2)=(3,-1)\)

\(\displaystyle =\frac{-1-7}{3-(-2)}=\frac{-8}{5}\)

Since the line goes through \(\displaystyle (0,5)\) we know that \(\displaystyle (0,5)\) is the y-intercept.  

Since we are looking for parallel lines, we need to write the equation of a line that has the same slope as the original, which is \(\displaystyle -6\).

Slope-intercept form equation is \(\displaystyle y=mx+b\), where \(\displaystyle m\) is the slope and \(\displaystyle b\) is the y-intercept.

Therefore,

\(\displaystyle y=-6x+5\).

Step 1: Find the Slope

\(\displaystyle Slope=\frac{y_2-y_1}{x_2-x_1}=\frac{-3-1}{0-4}=\frac{-4}{-4}=1\)

Step 2: Find the y-intercept

Use the slope and a point in the original y-intercept

\(\displaystyle y=mx+b\)

\(\displaystyle -3=1(0)+b\)

\(\displaystyle b=-3\)

Step 3: Write your equation

\(\displaystyle y=x-3\)

Since we know the slope and we know a point on the line we can use those two piece of information to find the y-intercept.

\(\displaystyle y=mx+b\)

\(\displaystyle 2=\frac{-3}{4}(8)+b\)

\(\displaystyle 2=-6+b\)

\(\displaystyle b=8\)

\(\displaystyle y=\frac{-3}{4}x+8\)

Slope is the change of a line. To find this line one can remember it as rise over run. This rise over run is really the change in the y direction over the change in the x direction.

Therefore the formula for slope is as follows.

\(\displaystyle Slope = \frac{\Delta Y}{\Delta X}=\frac{y_2-y_1}{x_2-x_1}\)

Plugging in our given points 

\(\displaystyle (x_1,y_1)=(-1,5)\) and \(\displaystyle (x_2,y_2)=(2,3)\),

\(\displaystyle Slope=\frac{y2-y1}{x2-x1}=\frac{5-3}{-1-2}=\frac{2}{-3}=-\frac{2}{3}\).

To find the equation of this line, first find the slope. Recall that slope is the change in y over the change in x: \(\displaystyle \frac{6-4}{2+1}=\frac{2}{3}\). Then, pick a point and use the slope to plug into the point-slope formula (\(\displaystyle y-y_{1}=m(x-x_{1})\)): \(\displaystyle y-6=\frac{2}{3}(x-2)\). Distribute and simplify so that you solve for y: \(\displaystyle y=\frac{2}{3}x+\frac{14}{3}\).

In \(\displaystyle y=mx+b\) form, where y = maximum heart rate and x = age, we can express the relationship as: 

\(\displaystyle y=-x+220\)

We are looking for a graph with a slope of -1 and a y-intercept of 220.

The slope is -1 because as you grow one year older, your maximum heart rate decreases by 1.