Consider that the slope-intercept form of the straight line with slope (m) and y-intercept (c) is given by,
![y=mx+c](https://tex.z-dn.net/?f=y%3Dmx%2Bc)
a.
Modify the given equation as,
![\begin{gathered} \frac{x}{3}+\frac{y}{2}=1 \\ \frac{y}{2}=-\frac{x}{3}+1 \\ y=-\frac{2}{3}x+2 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cfrac%7Bx%7D%7B3%7D%2B%5Cfrac%7By%7D%7B2%7D%3D1%20%5C%5C%20%5Cfrac%7By%7D%7B2%7D%3D-%5Cfrac%7Bx%7D%7B3%7D%2B1%20%5C%5C%20y%3D-%5Cfrac%7B2%7D%7B3%7Dx%2B2%20%5Cend%7Bgathered%7D)
Thus, the equation in slope-intercept form can be written as,
![y=-\frac{2}{3}x+2](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B2%7D%7B3%7Dx%2B2)
b.
Modify the given equation as,
![\begin{gathered} 4x-3y+2=0 \\ 3y=4x+2 \\ y=\frac{4}{3}x+\frac{2}{3} \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%204x-3y%2B2%3D0%20%5C%5C%203y%3D4x%2B2%20%5C%5C%20y%3D%5Cfrac%7B4%7D%7B3%7Dx%2B%5Cfrac%7B2%7D%7B3%7D%20%5Cend%7Bgathered%7D)
Thus, the equation in slope-intercept form can be written as,
![y=\frac{4}{3}x+\frac{2}{3}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B4%7D%7B3%7Dx%2B%5Cfrac%7B2%7D%7B3%7D)
c.
Modify the given equation as,
![\begin{gathered} x-y=5(x-y) \\ x-y=5x-5y \\ 5y-y=5x-x \\ 4y=4x \\ y=x \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20x-y%3D5%28x-y%29%20%5C%5C%20x-y%3D5x-5y%20%5C%5C%205y-y%3D5x-x%20%5C%5C%204y%3D4x%20%5C%5C%20y%3Dx%20%5Cend%7Bgathered%7D)
Thus, the equation in slope-intercept form can be written as,
Answer:
(a) Equation: ![T_n = 11.25 - 1.25n](https://tex.z-dn.net/?f=T_n%20%3D%2011.25%20-%201.25n)
(b) 7.5 gallons
Step-by-step explanation:
Given
Current Gas = 10 gallon
Every Hour = -1.25 gallon
Solving for (a): An equation for the scenario
This question represents an Arithmetic Progression and will be solved using the following formula:
![T_n = a + (n - 1)d](https://tex.z-dn.net/?f=T_n%20%3D%20a%20%2B%20%28n%20-%201%29d)
Where
![a = 10](https://tex.z-dn.net/?f=a%20%3D%2010)
![d = -1.25](https://tex.z-dn.net/?f=d%20%3D%20-1.25)
Substitute these values in the formula
![T_n = 10 + (n - 1) * -1.25](https://tex.z-dn.net/?f=T_n%20%3D%2010%20%2B%20%28n%20-%201%29%20%2A%20-1.25)
![T_n = 10 -1.25n +1.25](https://tex.z-dn.net/?f=T_n%20%3D%2010%20-1.25n%20%2B1.25)
Collect Like Terms
![T_n = 10 + 1.25 - 1.25n](https://tex.z-dn.net/?f=T_n%20%3D%2010%20%2B%201.25%20-%201.25n)
![T_n = 11.25 - 1.25n](https://tex.z-dn.net/?f=T_n%20%3D%2011.25%20-%201.25n)
Solving for (b): In this case, n - 3
Substitute 3 for n in ![T_n = 11.25 - 1.25n](https://tex.z-dn.net/?f=T_n%20%3D%2011.25%20-%201.25n)
![T_3 = 11.25 - 1.25 * 3](https://tex.z-dn.net/?f=T_3%20%3D%2011.25%20-%201.25%20%2A%203)
![T_3 = 11.25 - 3.75](https://tex.z-dn.net/?f=T_3%20%3D%2011.25%20-%203.75)
![T_3 = 7.5](https://tex.z-dn.net/?f=T_3%20%3D%207.5)
<em>7.5 gallon of gas left</em>