we know that
If two lines are parallel, then, their slopes are equal.
The formula to calculate the slope between two points is equal to
![m=\frac{y2-y1}{x2-x1}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7By2-y1%7D%7Bx2-x1%7D)
we will proceed to calculate the slope in each case, to determine the solution of the problem
<u>Case A)</u> Point ![B(5,2)\ C(7,-5)](https://tex.z-dn.net/?f=B%285%2C2%29%5C%20C%287%2C-5%29)
Find the slope BC
Substitute the values in the formula
![m=\frac{-5-2}{7-5}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B-5-2%7D%7B7-5%7D)
![m=\frac{-7}{2}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B-7%7D%7B2%7D)
![m=-3.5](https://tex.z-dn.net/?f=m%3D-3.5)
so
The equation
is parallel to the line passing through the points ![B(5,2)\ C(7,-5)](https://tex.z-dn.net/?f=B%285%2C2%29%5C%20C%287%2C-5%29)
therefore
<u>the answer Part A) is</u>
------> ![y=-3.5x-15](https://tex.z-dn.net/?f=y%3D-3.5x-15)
<u>Case B)</u> Point ![D(11,6)\ E(5,9)](https://tex.z-dn.net/?f=D%2811%2C6%29%5C%20E%285%2C9%29)
Find the slope DE
Substitute the values in the formula
![m=\frac{9-6}{5-11}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B9-6%7D%7B5-11%7D)
![m=\frac{3}{-6}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B3%7D%7B-6%7D)
![m=-0.5](https://tex.z-dn.net/?f=m%3D-0.5)
so
The equation
is parallel to the line passing through the points ![D(11,6)\ E(5,9)](https://tex.z-dn.net/?f=D%2811%2C6%29%5C%20E%285%2C9%29)
therefore
<u>the answer Part B) is</u>
------> ![y=-0.5x-3](https://tex.z-dn.net/?f=y%3D-0.5x-3)
<u>Case C)</u> Point ![F(-7,12)\ G(3,-8)](https://tex.z-dn.net/?f=F%28-7%2C12%29%5C%20G%283%2C-8%29)
Find the slope FG
Substitute the values in the formula
![m=\frac{-20}{10}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B-20%7D%7B10%7D)
![m=-2](https://tex.z-dn.net/?f=m%3D-2)
so
Any linear equation with slope
will be parallel to the line passing through the points ![F(-7,12)\ G(3,-8)](https://tex.z-dn.net/?f=F%28-7%2C12%29%5C%20G%283%2C-8%29)
<u>Case D)</u> Point ![H(4,4)\ I(8,9)](https://tex.z-dn.net/?f=H%284%2C4%29%5C%20I%288%2C9%29)
Find the slope HI
Substitute the values in the formula
![m=\frac{9-4}{8-4}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B9-4%7D%7B8-4%7D)
![m=\frac{5}{4}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B5%7D%7B4%7D)
![m=1.25](https://tex.z-dn.net/?f=m%3D1.25)
so
The equation
is parallel to the line passing through the points ![H(4,4)\ I(8,9)](https://tex.z-dn.net/?f=H%284%2C4%29%5C%20I%288%2C9%29)
therefore
<u>the answer Part D) is</u>
------> ![y=1.25x+4](https://tex.z-dn.net/?f=y%3D1.25x%2B4)
<u>Case E)</u> Point ![J(7,2)\ K(-9,8)](https://tex.z-dn.net/?f=J%287%2C2%29%5C%20K%28-9%2C8%29)
Find the slope JK
Substitute the values in the formula
![m=\frac{8-2}{-9-7}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B8-2%7D%7B-9-7%7D)
![m=\frac{6}{-16}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B6%7D%7B-16%7D)
![m=-0.375](https://tex.z-dn.net/?f=m%3D-0.375)
so
Any linear equation with slope
will be parallel to the line passing through the points ![J(7,2)\ K(-9,8)](https://tex.z-dn.net/?f=J%287%2C2%29%5C%20K%28-9%2C8%29)
<u>Case F)</u> Point ![L(5,-7)\ M(4,-12)](https://tex.z-dn.net/?f=L%285%2C-7%29%5C%20M%284%2C-12%29)
Find the slope LM
Substitute the values in the formula
![m=\frac{-12+7}{4-5}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B-12%2B7%7D%7B4-5%7D)
![m=\frac{-5}{-1}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B-5%7D%7B-1%7D)
![m=5](https://tex.z-dn.net/?f=m%3D5)
so
The equation
is parallel to the line passing through the points ![L(5,-7)\ M(4,-12)](https://tex.z-dn.net/?f=L%285%2C-7%29%5C%20M%284%2C-12%29)
therefore
<u>the answer Part F) is</u>
------> ![y=5x+19](https://tex.z-dn.net/?f=y%3D5x%2B19)