You should recognize that the given equation is in the point-slope form.

In order to find the y-intercept, rearrange the equation into slope-intercept form, \(\displaystyle y=mx+b\).

\(\displaystyle y-8=\frac{1}{10}(x+200)\)

\(\displaystyle y-8=\frac{1}{10}x+20\)

\(\displaystyle y=\frac{1}{10}x+28\)

Since, \(\displaystyle b=28\), the y-intercept must be located at \(\displaystyle (0,28)\).

You should recognize that the given equation is in the point-slope form.

In order to find the y-intercept, rearrange the equation into slope-intercept form, \(\displaystyle y=mx+b\).

\(\displaystyle y-7=-2(x-10)\)

\(\displaystyle y-7=-2x+20\)

\(\displaystyle y=-2x+27\)

Since, \(\displaystyle b=27\), the y-intercept must be located at \(\displaystyle (0,27)\).

You should recognize that the given equation is in the point-slope form.

In order to find the y-intercept, rearrange the equation into slope-intercept form, \(\displaystyle y=mx+b\).

\(\displaystyle y+15=-7(x-2)\)

\(\displaystyle y+15=-7x+14\)

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

Since, \(\displaystyle b=-1\), the y-intercept must be located at \(\displaystyle (0,-1)\).

You should recognize that the given equation is in the point-slope form.

In order to find the y-intercept, rearrange the equation into slope-intercept form, \(\displaystyle y=mx+b\).

\(\displaystyle y-24=-8(x+1)\)

\(\displaystyle y-24=-8x-8\)

\(\displaystyle y=-8x+16\)

Since, \(\displaystyle b=16\), the y-intercept must be located at \(\displaystyle (0,16)\).

You should recognize that the given equation is in the point-slope form.

In order to find the y-intercept, rearrange the equation into slope-intercept form, \(\displaystyle y=mx+b\).

\(\displaystyle y+29=-\frac{1}{4}(x+36)\)

\(\displaystyle y+29=-\frac{1}{4}x-9\)

\(\displaystyle y=-\frac{1}{4}x-38\)

Since, \(\displaystyle b=-38\), the y-intercept must be located at \(\displaystyle (0,-38)\).

You should recognize that the given equation is in the point-slope form.

In order to find the y-intercept, rearrange the equation into slope-intercept form, \(\displaystyle y=mx+b\).

\(\displaystyle y-10=-20(x+\frac{1}{4})\)

\(\displaystyle y-10=-20x-5\)

\(\displaystyle y=-20x+5\)

Since, \(\displaystyle b=5\), the y-intercept must be located at \(\displaystyle (0,5)\).

The \(\displaystyle y\)-intercept of a line is the point at which it intersects the \(\displaystyle y\)-axis; its \(\displaystyle x\)-coordinate at this point is 0, so its \(\displaystyle y\)-coordinate can be found by substituting 0 for \(\displaystyle x\) in the equation and solving for \(\displaystyle y\). Do this with both equations:

\(\displaystyle 2x+y = 12\)

\(\displaystyle 2 \cdot 0 +y = 12\)

\(\displaystyle 0+ y = 12\)

\(\displaystyle y = 12\)

The \(\displaystyle y\)-intercept of the line is at \(\displaystyle (0, 12)\).

\(\displaystyle x+2y = 12\)

\(\displaystyle 0+2y = 12\)

\(\displaystyle 2y = 12\)

\(\displaystyle \frac{2y }{2}= \frac{12}{2}\)

\(\displaystyle y = 6\)

The \(\displaystyle y\)-intercept of the line is at \(\displaystyle (0, 6)\).

The two lines have different \(\displaystyle y\)-intercepts.

The \(\displaystyle y\)-intercept of a line is the point at which it intersects the \(\displaystyle y\)-axis; its \(\displaystyle x\)-coordinate at this point is 0.

The \(\displaystyle y\)-coordinate of the line of the equation

\(\displaystyle x+ y = 6\)

can be found by substituting 0 for \(\displaystyle x\) in the equation and solving for \(\displaystyle y\):

\(\displaystyle 0+ y = 6\)

\(\displaystyle y=6\)

The \(\displaystyle y\)-intercept of this line is at \(\displaystyle (0,6)\).

The line with equation \(\displaystyle y = b\) is a horizontal line with its \(\displaystyle y\)-intercept at \(\displaystyle (0,b)\), so the \(\displaystyle y\)-intercept line of the equation \(\displaystyle y=6\) has its \(\displaystyle y\)-intercept at \(\displaystyle (0,6)\) as well.

Both lines indeed have the same \(\displaystyle y\)-intercept.

Both equations are given in the slope-intercept form \(\displaystyle y = mx+b\), in which the stand-alone constant \(\displaystyle b\) is the \(\displaystyle y\)-coordinate of the \(\displaystyle y\)-intercept. In both equations, this value is 117, so \(\displaystyle (0, 117 )\) is the \(\displaystyle y\)-intercept of both equations.

Recall the point-slope form of the equation of a line that has a slope of \(\displaystyle m\) and passes through the point \(\displaystyle (x_1, y_1)\):

\(\displaystyle y-y_1=m(x-x_1)\)

Plug in the given point and the given slope.

\(\displaystyle y-4=5(x-12)\)

Rearrange the equation into slope-intercept form.

\(\displaystyle y-4=5x-60\)

\(\displaystyle y=5x-56\)

The y-intercept for this line is \(\displaystyle (0, -56)\).