The angles are supplementary, therefore, the sum of the angles must equal \(\displaystyle 180^o\).

\(\displaystyle \small (4n+22^o)+(8n-10^o)=180^o\)

\(\displaystyle \small 4n+22^o+8n-10^o=180^o\)

\(\displaystyle \small 12n+12^o=180^o\)

\(\displaystyle \small 12n=168^o\)

\(\displaystyle \small n=14^o\)

Since supplementary angles must add up to \(\displaystyle 180^{\circ}\), the given angles are indeed supplementary.

The angles containing the variable \(\displaystyle x\) all reside along one line, therefore, their sum must be \(\displaystyle 180^o\).

\(\displaystyle \small 5x+4x+3x=180^o\)

\(\displaystyle \small 12x=180^o\)

\(\displaystyle \small x=15^o\)

Because \(\displaystyle 2y\) and \(\displaystyle 5x\) are opposite angles, they must be equal.

\(\displaystyle \small 2y=5x\)

\(\displaystyle \small x=15^o\)

\(\displaystyle \small 2y=5(15^o)=75^o\)

\(\displaystyle \small y=\frac{75^o}{2}=37.5^o\)

Which of the following could be the function modeled by this graph?

We can begin here by trying to identify a couple points  on the graph

We can see that it crosses the y-axis at \(\displaystyle (0,-7)\)

Therefore, not only do we have a point, we have the y-intercept. This tells us that the equation of the line needs to have a \(\displaystyle -7\) in it somewhere. Eliminate any option that do not have this feature.

Next, find the slope by counting up and over from the y-intercept to the next clear point.

It seems like the line goes up 5 and right 1 to the point \(\displaystyle (1,-2)\)

This means we have a slope of 5, which means our equation must look like this:

\(\displaystyle y=5x-7\)

For the linear function in point-slope form

\(\displaystyle y-y_{1}=m\left ( x-x_{1}\right )\)

The slope is equal to \(\displaystyle m.\)

For this problem

\(\displaystyle y-3=5(x+2)\)

we get

\(\displaystyle m=5\)

For the linear function in point-slope form

\(\displaystyle y-y_{1}=m\left ( x-x_{1}\right )\)

The slope is equal to \(\displaystyle m.\)

For this problem

\(\displaystyle y-7=10(x+1)\)

we get

\(\displaystyle m=10\)

By definition, the y-intercept is the point on the line that crosses the y-axis. This can be found by substituting \(\displaystyle x = 0\) into the equation. When we do this with our equation, 

\(\displaystyle y = 3x + 4 = 3\cdot0 + 4 = 0 + 4 = 4\). 

Alternatively, you can remember \(\displaystyle y = mx + b\) form, a general form for a line in which \(\displaystyle m\) is the slope and \(\displaystyle b\) is the y-intercept. 

Recall slope-intercept form, or \(\displaystyle y = mx + b\). In this form, \(\displaystyle m\) is the slope and \(\displaystyle b\) is the y-intercept. Given our equation above, the slope must be the coefficient of the x, which is \(\displaystyle \frac{2}{3}\). 

The x-intercept of an equation is the point at which the line crosses the x-axis. Thus, we can find the x-intercept by plugging in \(\displaystyle y=0\). When we do this with our equation: 

\(\displaystyle y = 3x - 6 \rightarrow 0 = 3x - 6\rightarrow6 = 3x \rightarrow2 = x\)

Thus, our x-intercept is the point \(\displaystyle (2,0)\). 

First, we need to compute \(\displaystyle m\), the slope. We can do this with the slope formula

\(\displaystyle m = \frac{y_2-y_1}{x_2-x_1}\), sometimes called "rise over run"

\(\displaystyle m = \frac{7-2}{4-0} = \frac{5}{4}\)

 So we now have

\(\displaystyle y = \frac{5}{4}x +b\)

Now in order to solve for \(\displaystyle b,\) we substitute one of our points into the equation we found. It doesn't matter which point we use, so we'll use \(\displaystyle (0,2)\).

We then have:

\(\displaystyle 2 = \frac{5}{4}0 +b\)

Which becomes \(\displaystyle 2 = b\).

Hence we take our found value for \(\displaystyle b\) and plug it back into \(\displaystyle y= \frac{5}{4}x+b\) to get

\(\displaystyle y =\frac{5}{4}x+2\).