<u>Part 1)</u>
we know that
the equation of the line in slope-intercept form is equal to
![y=mx+b](https://tex.z-dn.net/?f=y%3Dmx%2Bb)
where
m is the slope
b is the y-intercept
we have
![2x-3y=9](https://tex.z-dn.net/?f=2x-3y%3D9)
solve for y
![3y=2x-9](https://tex.z-dn.net/?f=3y%3D2x-9)
-------> equation of the line in slope-intercept form
so
the slope m is ![\frac{2}{3}](https://tex.z-dn.net/?f=%5Cfrac%7B2%7D%7B3%7D)
the y-intercept b is ![-3](https://tex.z-dn.net/?f=-3)
<u>Part 2)</u>
we know that
the equation of the line in slope-intercept form is equal to
![y=mx+b](https://tex.z-dn.net/?f=y%3Dmx%2Bb)
where
m is the slope
b is the y-intercept
we have
![x-4y=-20](https://tex.z-dn.net/?f=x-4y%3D-20)
solve for y
![4y=x+20](https://tex.z-dn.net/?f=4y%3Dx%2B20)
-------> equation of the line in slope-intercept form
so
the slope m is ![\frac{1}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B4%7D)
the y-intercept b is ![5](https://tex.z-dn.net/?f=5)
<u>Part 3)</u>
we know that
The x-intercept is the value of x when the value of y is equal to zero
The y-intercept is the value of y when the value of x is equal to zero
we have
![-x+4y=12](https://tex.z-dn.net/?f=-x%2B4y%3D12)
a) Find the x-intercept
For
substitute in the equation
![-x+4*0=12](https://tex.z-dn.net/?f=-x%2B4%2A0%3D12)
![x=-12](https://tex.z-dn.net/?f=x%3D-12)
The answer part 3a) is ![(-12,0)](https://tex.z-dn.net/?f=%28-12%2C0%29)
b) Find the y-intercept
For
substitute in the equation
![-0+4y=12](https://tex.z-dn.net/?f=-0%2B4y%3D12)
![y=3](https://tex.z-dn.net/?f=y%3D3)
The answer part 3b) is ![(0,3)](https://tex.z-dn.net/?f=%280%2C3%29)
<u>Part 4)</u>
we know that
the equation of the line in standard form is
we have
![y=\frac{2}{3}x+7](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B2%7D%7B3%7Dx%2B7)
Multiply by
both sides
![3y=2x+21](https://tex.z-dn.net/?f=3y%3D2x%2B21)
------> equation in standard form
therefore
the answer Part 4) is option B False
<u>Part 5)</u>
Step 1
<u>Find the slope</u>
we have
![2x-5y=12](https://tex.z-dn.net/?f=2x-5y%3D12)
solve for y
![5y=2x-12](https://tex.z-dn.net/?f=5y%3D2x-12)
![y=(2/5)x-(12/5)](https://tex.z-dn.net/?f=y%3D%282%2F5%29x-%2812%2F5%29)
so
the slope m is ![\frac{2}{5}](https://tex.z-dn.net/?f=%5Cfrac%7B2%7D%7B5%7D)
Step 2
Find the y-intercept
The y-intercept is the value of y when the value of x is equal to zero
we have
![4y+24=5x](https://tex.z-dn.net/?f=4y%2B24%3D5x)
for ![x=0](https://tex.z-dn.net/?f=x%3D0)
![4y+24=5*0](https://tex.z-dn.net/?f=4y%2B24%3D5%2A0)
![4y=-24](https://tex.z-dn.net/?f=4y%3D-24)
![y=-6](https://tex.z-dn.net/?f=y%3D-6)
the y-intercept is ![-6](https://tex.z-dn.net/?f=-6)
Step 3
Find the equation of the line
we have
![m=\frac{2}{5}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B2%7D%7B5%7D)
![b=-6](https://tex.z-dn.net/?f=b%3D-6)
the equation of the line in slope-intercept form is
![y=mx+b](https://tex.z-dn.net/?f=y%3Dmx%2Bb)
substitute the values
![y=\frac{2}{5}x-6](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B2%7D%7B5%7Dx-6)
therefore
the answer Part 5) is the option A ![y=\frac{2}{5}x-6](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B2%7D%7B5%7Dx-6)
<u>Part 6) </u>
Step 1
<u>Find the slope of the given line</u>
we know that
if two lines are perpendicular. then the product of their slopes is equal to minus one
so
![m1*m2=-1](https://tex.z-dn.net/?f=m1%2Am2%3D-1)
in this problem
the given line
![x+8y=27](https://tex.z-dn.net/?f=x%2B8y%3D27)
solve for y
![8y=27-x](https://tex.z-dn.net/?f=8y%3D27-x)
![y=(27/8)-(x/8)](https://tex.z-dn.net/?f=y%3D%2827%2F8%29-%28x%2F8%29)
the slope m1 is ![m1=-\frac{1}{8}](https://tex.z-dn.net/?f=m1%3D-%5Cfrac%7B1%7D%7B8%7D)
so
the slope m2 is ![m2=8](https://tex.z-dn.net/?f=m2%3D8)
Step 2
<u>Find the equation of the line</u>
we know that
the equation of the line in slope point form is equal to
![y-y1=m*(x-x1)](https://tex.z-dn.net/?f=y-y1%3Dm%2A%28x-x1%29)
we have
![m2=8](https://tex.z-dn.net/?f=m2%3D8)
point ![(-5,5)](https://tex.z-dn.net/?f=%28-5%2C5%29)
substitutes the values
![y-5=8*(x+5)](https://tex.z-dn.net/?f=y-5%3D8%2A%28x%2B5%29)
![y=8x+40+5](https://tex.z-dn.net/?f=y%3D8x%2B40%2B5)
![y=8x+45](https://tex.z-dn.net/?f=y%3D8x%2B45)
therefore
the answer part 6) is the option C ![y=8x+45](https://tex.z-dn.net/?f=y%3D8x%2B45)
<u>Part 7)</u>
-------> the slope is ![m=(8/3)](https://tex.z-dn.net/?f=m%3D%288%2F3%29)
![8x- y=17](https://tex.z-dn.net/?f=8x-%20y%3D17)
--------> the slope is ![m=8](https://tex.z-dn.net/?f=m%3D8)
we know that
if two lines are parallel , then their slopes are the same
in this problem the slopes are not the same
therefore
the answer part 7) is the option D) No, since the slopes are different.
<u>Part 8)</u>
a. Write an equation for the line in point-slope form
b. Rewrite the equation in standard form using integers
Step 1
<u>Find the slope of the line</u>
we know that
the slope between two points is equal to
![m=\frac{(y2-y1)}{(x2-x1)}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B%28y2-y1%29%7D%7B%28x2-x1%29%7D)
substitute the values
![m=\frac{(4+1)}{(8-2)}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B%284%2B1%29%7D%7B%288-2%29%7D)
Step 2
<u>Find the equation in point slope form</u>
we know that
the equation of the line in slope point form is equal to
![y-y1=m*(x-x1)](https://tex.z-dn.net/?f=y-y1%3Dm%2A%28x-x1%29)
we have
![m=(5/6)](https://tex.z-dn.net/?f=m%3D%285%2F6%29)
point ![(2,-1)](https://tex.z-dn.net/?f=%282%2C-1%29)
substitutes the values
-------> equation of the line in point slope form
Step 3
<u>Rewrite the equation in standard form using integers</u>
![y=(5/6)x-(5/3)-1](https://tex.z-dn.net/?f=y%3D%285%2F6%29x-%285%2F3%29-1)
![y=(5/6)x-(8/3)](https://tex.z-dn.net/?f=y%3D%285%2F6%29x-%288%2F3%29)
Multiply by
both sides
![6y=5x-16](https://tex.z-dn.net/?f=6y%3D5x-16)
--------> equation of the line in standard form
<u>Part 9)</u>
we know that
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%7B%28y2-y1%29%7D%7B%28x2-x1%29%7D)
where
(x1,y1) ------> is the first point
(x2,y2) -----> is the second point
In the numerator calculate the difference of the y-coordinates
in the denominator calculate the difference of the x-coordinates
<u>Part 10)</u>
we know that
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%7B%28y2-y1%29%7D%7B%28x2-x1%29%7D)
substitutes
![m=\frac{(5+1)}{(-1+3)}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B%285%2B1%29%7D%7B%28-1%2B3%29%7D)
![m=\frac{(6)}{(2)}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B%286%29%7D%7B%282%29%7D)
![m=3](https://tex.z-dn.net/?f=m%3D3)
therefore
the answer Part 10) is ![m=3](https://tex.z-dn.net/?f=m%3D3)
<u>Part 11)</u>
we know that
the equation of the line in slope point form is equal to
![y-y1=m*(x-x1)](https://tex.z-dn.net/?f=y-y1%3Dm%2A%28x-x1%29)
substitute the values
--------> this is the equation in the point slope form