Question 4
Recall the following facts:
1. If two lines are parallel, their slopes are the same
2. If two lines are perpendicular, the product of their slopes equals 1.
The line given to us is:
![y = \frac{3}{4} x + 12](https://tex.z-dn.net/?f=y%20%3D%20%20%5Cfrac%7B3%7D%7B4%7D%20x%20%2B%2012)
By comparing to ;
![y = mx + c](https://tex.z-dn.net/?f=y%20%3D%20mx%20%2B%20c)
The slope is
![m = \frac{3}{4}](https://tex.z-dn.net/?f=m%20%3D%20%20%5Cfrac%7B3%7D%7B4%7D%20)
Now let us compare this to:
![y = \frac{4}{3} x - 2](https://tex.z-dn.net/?f=y%20%3D%20%20%5Cfrac%7B4%7D%7B3%7D%20x%20-%202)
which has slope
![= \frac{4}{3}](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B4%7D%7B3%7D%20)
Now let us check to see if the two are parallel,
![\frac{3}{ 4} \neq \frac{4}{3}](https://tex.z-dn.net/?f=%20%5Cfrac%7B3%7D%7B%204%7D%20%20%20%5Cneq%20%5Cfrac%7B4%7D%7B3%7D%20)
Since the two slopes are not the same, they are not parallel.
Now let us check to see if they are perpendicular
![\frac{3}{4} \times \frac{4}{3} \neq - 1](https://tex.z-dn.net/?f=%20%5Cfrac%7B3%7D%7B4%7D%20%20%5Ctimes%20%20%5Cfrac%7B4%7D%7B3%7D%20%20%5Cneq%20-%201)
Since their product is not -1, the two lines are not perpendicular.
Hence ,
![y = \frac{3}{4} x + 12](https://tex.z-dn.net/?f=y%20%3D%20%20%5Cfrac%7B3%7D%7B4%7D%20x%20%2B%2012)
is neither parallel or perpendicular to
![y = \frac{4}{3} x - 2](https://tex.z-dn.net/?f=y%20%3D%20%20%5Cfrac%7B4%7D%7B3%7D%20x%20-%202)
The next equation is
![y = - \frac{4}{3} x + 5](https://tex.z-dn.net/?f=y%20%3D%20%20-%20%20%5Cfrac%7B4%7D%7B3%7D%20x%20%2B%205)
The slope of this equation is
![= - \frac{4}{3}](https://tex.z-dn.net/?f=%20%3D%20%20-%20%20%5Cfrac%7B4%7D%7B3%7D%20)
Since,
![\frac{3}{4} \times - \frac{4}{3} = - 1](https://tex.z-dn.net/?f=%20%5Cfrac%7B3%7D%7B4%7D%20%20%5Ctimes%20%20-%20%20%5Cfrac%7B4%7D%7B3%7D%20%20%3D%20%20-%201)
The equation
![y = \frac{3}{4} x + 12](https://tex.z-dn.net/?f=y%20%3D%20%20%5Cfrac%7B3%7D%7B4%7D%20x%20%2B%2012)
is perpendicular to
![y = - \frac{4}{3} x + 5](https://tex.z-dn.net/?f=y%20%3D%20%20-%20%20%5Cfrac%7B4%7D%7B3%7D%20x%20%2B%205)
The next equation is
![y = \frac{3}{4} x](https://tex.z-dn.net/?f=y%20%3D%20%20%5Cfrac%7B3%7D%7B4%7D%20x)
Since the slope if the two are equal, that is
![\frac{3}{4} = \frac{3}{4}](https://tex.z-dn.net/?f=%20%5Cfrac%7B3%7D%7B4%7D%20%20%3D%20%20%5Cfrac%7B3%7D%7B4%7D%20)
the two equations are parallel.
The next equation is
![y = - \frac{4}{3} x - 6](https://tex.z-dn.net/?f=y%20%3D%20%20-%20%20%5Cfrac%7B4%7D%7B3%7D%20x%20%20-%206)
Since
![\frac{ 3}{4} \times \frac{ - 4}{3} = -1](https://tex.z-dn.net/?f=%20%20%5Cfrac%7B%203%7D%7B4%7D%20%5Ctimes%20%20%5Cfrac%7B%20-%204%7D%7B3%7D%20%20%20%3D%20%20-1)
the two equations are perpendicular.
Question 5.
The given line in the graph passes through,
(10,7), (-8,-5) and (1,1).
Using any two points we can determine the slope,
![= \frac{7 - - 5}{10 - - 8}](https://tex.z-dn.net/?f=%20%3D%20%20%5Cfrac%7B7%20-%20%20-%205%7D%7B10%20-%20%20-%208%7D%20)
![= \frac{7 +5 }{10 + 8}](https://tex.z-dn.net/?f=%20%3D%20%20%5Cfrac%7B7%20%2B5%20%7D%7B10%20%2B%208%7D%20)
![= \frac{12}{18} = \frac{2}{3}](https://tex.z-dn.net/?f=%20%3D%20%20%5Cfrac%7B12%7D%7B18%7D%20%20%3D%20%20%5Cfrac%7B2%7D%7B3%7D%20)
The line parallel to this line which passes through (5,-1), also has slope
![= \frac{2}{3}](https://tex.z-dn.net/?f=%20%3D%20%20%5Cfrac%7B2%7D%7B3%7D%20)
The equation of this line is
![y - - 1 = \frac{2}{3} (x - 5)](https://tex.z-dn.net/?f=y%20-%20%20-%201%20%3D%20%20%5Cfrac%7B2%7D%7B3%7D%20%28x%20-%205%29)
This implies that,
![y + 1 = \frac{2}{3} x - \frac{10}{3}](https://tex.z-dn.net/?f=y%20%2B%201%20%3D%20%20%5Cfrac%7B2%7D%7B3%7D%20x%20-%20%20%5Cfrac%7B10%7D%7B3%7D%20)
This simplifies to
![y = \frac{2}{3} x - \frac{13}{3}](https://tex.z-dn.net/?f=y%20%3D%20%20%5Cfrac%7B2%7D%7B3%7D%20x%20-%20%20%5Cfrac%7B13%7D%7B3%7D%20)
To find any point on this line, choose any value for x and solve the corresponding y value. So when
![x = 2](https://tex.z-dn.net/?f=x%20%3D%202)
![y = \frac{2}{3} \times 2 - \frac{13}{3} = - 3](https://tex.z-dn.net/?f=y%20%3D%20%20%5Cfrac%7B2%7D%7B3%7D%20%20%5Ctimes%202%20-%20%20%5Cfrac%7B13%7D%7B3%7D%20%20%3D%20%20-%203)
Hence
![(2 \: \: - 3)](https://tex.z-dn.net/?f=%282%20%5C%3A%20%20%5C%3A%20%20-%203%29)
is a point on this line.