Any line that is parallel to a line \(\displaystyle y=mx+b\) has a slope that is equal to the slope \(\displaystyle m\). Given \(\displaystyle y=\frac{1}{3}x+9\), \(\displaystyle m=\frac{1}{3}\) and therefore any line parallel to the given line must have a slope of \(\displaystyle \frac{1}{3}\).

Any line that is parallel to a line \(\displaystyle y=mx+b\) has a slope that is equal to the slope \(\displaystyle m\). Given \(\displaystyle y=-5x+11\), \(\displaystyle m=-5\) and therefore any line parallel to the given line must have a slope of \(\displaystyle -5\).

Any line that is parallel to a line \(\displaystyle y=mx+b\) has a slope that is equal to the slope \(\displaystyle m\). Given \(\displaystyle y=\frac{4}{3}x+12\), \(\displaystyle m=\frac{4}{3}\) and therefore any line parallel to the given line must have a slope of \(\displaystyle \frac{4}{3}\).

Parallel lines have the same slope. Therefore, the slope of the new line is \(\displaystyle 2\), as the equation of the original line is \(\displaystyle y=2x+10\) \(\displaystyle (y=mx+b)\),with slope \(\displaystyle m\).

\(\displaystyle m=2\)     and      \(\displaystyle (5,1)\):

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

\(\displaystyle y-1=2(x-5)\)

\(\displaystyle y-1=2x-10\)

\(\displaystyle y=2x-10+1\)

\(\displaystyle y=2x-9\)

The parallel line has the equation \(\displaystyle \dpi{100} \small 4x-2y=5\). We can find the slope by putting the equation into slope-intercept form, y = mx + b, where m is the slope and b is the intercept.  \(\displaystyle \dpi{100} \small 4x-2y=5\) becomes \(\displaystyle \dpi{100} \small y=2x-\frac{5}{2}\), so the slope is 2.

We know that our line must have an equation that looks like \(\displaystyle \dpi{100} \small y=2x+b\). Now we need the intercept. We can solve for b by plugging in the point (4, 1).

1 = 2(4) + b

b = –7

Then the line in question is \(\displaystyle \dpi{100} \small y=2x-7\).

Parallel lines have the same slope, so our slope will still be 4. The y-intercept is just the "+b" at the end. In f(x) the y-intercept is 13. In this case, we need to have a y-intercept of -13, so our equation just becomes:

\(\displaystyle \small \small f(x)=4x-13\)

Two lines are parallel if they have the same slope. The slope of g(x) is 6, so eliminate anything without a slope of 6.

Recall slope intercept form which is \(\displaystyle y=mx+b\).

We know that the line must have an m of 6 and an (x,y) of (8,9). Plug everything in and go from there.

\(\displaystyle 9=(6\cdot 8)+b\)

\(\displaystyle \small 9=48+b\)

\(\displaystyle \small b=9-48=-39\)

So we get:

\(\displaystyle \small y=6x-39\)

Given a line \(\displaystyle a\) defined by the equation \(\displaystyle f(x)=mx+b\) with slope \(\displaystyle m\), any line that is parallel to \(\displaystyle a\) also has a slope of \(\displaystyle m\). Since \(\displaystyle f(x)=7x-12\), the slope \(\displaystyle m\) is \(\displaystyle 7\) and the slope of any line \(\displaystyle g(x)\) parallel to \(\displaystyle f(x)\) also has a slope of \(\displaystyle m=7\).

Since \(\displaystyle g(x)\) also needs to have a \(\displaystyle y\)-intercept of \(\displaystyle 8\), then the equation for \(\displaystyle g(x)\) must be \(\displaystyle g(x)=7x+8\). 

Given a line \(\displaystyle a\) defined by the equation \(\displaystyle f(x)=mx+b\) with slope \(\displaystyle m\), any line that is parallel to \(\displaystyle a\) also has a slope of \(\displaystyle m\). Since \(\displaystyle f(x)=-\frac{6}{5}x+7\), the slope \(\displaystyle m\) is \(\displaystyle -\frac{6}{5}\) and the slope of any line \(\displaystyle g(x)\) parallel to \(\displaystyle f(x)\) also has a slope of \(\displaystyle m=-\frac{6}{5}\).

Since \(\displaystyle g(x)\) also needs to have a \(\displaystyle y\)-intercept of \(\displaystyle -2\), then the equation for \(\displaystyle g(x)\) must be \(\displaystyle g(x)=-\frac{6}{5}x-2\). 

Given a line \(\displaystyle a\) defined by the equation \(\displaystyle f(x)=mx+b\) with slope \(\displaystyle m\), any line that is parallel to \(\displaystyle a\) also has a slope of \(\displaystyle m\). Since \(\displaystyle f(x)=2x+5\), the slope \(\displaystyle m\) is \(\displaystyle 2\) and the slope of any line \(\displaystyle g(x)\) parallel to \(\displaystyle f(x)\) also has a slope of \(\displaystyle m=2\).

Since \(\displaystyle g(x)\) also needs to have a \(\displaystyle y\)-intercept of \(\displaystyle -17\), then the equation for \(\displaystyle g(x)\) must be \(\displaystyle g(x)=2x-17\).