Answer:
1. [tex]y=\frac{1}{5}x-\frac{32}{5}[/tex]
2. [tex]y=-5x+30[/tex]
Step-by-step explanation:
Preliminary Material:
A line that is perpendicular to another has the opposite reciprocal slope of the other line.
- Ex: [tex]-\frac{1}{3}x+3[/tex] is perpendicular to [tex]3x+2[/tex] because [tex]-\frac{1}{3}[/tex] is an opposite reciprocal to [tex]3[/tex].
A line parallel to another has the same slope.
With this in mind, we can begin to answer this question.
1.
Setup for question 1:
A line perpendicular with [tex]-5x+5[/tex] that goes through the point (7, 5).
Before going over the point this perpendicular line must go through, let's make an equation with the slope of the perpendicular line.
A linear function can be represented with this formula: [tex]y=mx+b[/tex].
- [tex]m[/tex] = the slope of the line.
- b = the y intercept.
We know the slope of this function must be an opposite reciprocal to -5, and using the information I first gave in the beginning of this problem, we can have the slope of the function.
- the opposite reciprocal of [tex]-5[/tex] (or [tex]-\frac{5}{1}[/tex]) is [tex]\frac{1}{5}[/tex].
If we take this slope and insert it into the linear equation, we will have the function...
- [tex]y=\frac{1}{5}x+b[/tex]
So we now know the slope of this function, but what about the y intercept, [tex]b[/tex]? This is where the point given to us in the beginning of this problem comes into play. If we plug the point (7, -5) into our x and y coordinates in the above equation, we can solve for [tex]b[/tex].
Algebra:
[tex]-5=\frac{1}{5}(7)+b[/tex]
[tex]-5=\frac{7}{5}+b[/tex]
- [tex]-5[/tex] is the same as the fraction [tex]-\frac{25}{5}[/tex].
[tex]-\frac{25}{5}=\frac{7}{5}+b[/tex]
- subtract [tex]\frac{7}{5}[/tex] from both sides.
[tex]-\frac{32}{5}=b[/tex]
- [tex]b[/tex] is equal to [tex]-\frac{32}{5}[/tex] or [tex]-6\frac{2}{5}[/tex]. Lets plug this into our original equation, and we're done.
Answer:
Our final answer is: [tex]y=\frac{1}{5}x-\frac{32}{5}[/tex].
2.
There will be no setup for 2. I'll show the algebra without explanation as well. If there's any confusion you can ask me :).
Algebra:
[tex]y=mx+b[/tex]
[tex]y=-5x+b[/tex]
[tex]-5=-5(7)+b[/tex]
[tex]-5=-35+b[/tex]
[tex]b=30[/tex]
Answer:
- Our final answer is: [tex]y=-5x+30[/tex]
