Question 183218
Let x=price per barrel of oil and y=price per gallon of gasoline



First translation: "When oil was selling for $30 per barrel I paid $1.25 per gallon for gasoline" means that when {{{x=130}}}, {{{y=1.25}}}. So we have one point (130,1.25)



Second translation: "When oil was selling for $140 per barrel, I paid $4.00 for a gallon of gasoline" means that when {{{x=140}}}, {{{y=4}}}. So we have another point (140,4)



So here's the table of the two ordered pairs (points)
<table border=1>
<th>x</th><th>y</th>
<tr><td>130</td><td>1.25</td></tr>
<tr><td>140</td><td>4</td></tr>
</table>


Now set up the axis (with the proper ranges and labels)


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/step1-1.png">



Plot the two points


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/step2-1.png">


Draw a line through the points (in blue)


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/step3.png">


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Now let's find the equation of the line that goes through the points (130,1.25) and (140,4)




First let's find the slope of the line through the points (130,1.25) and (140,4)



Note: *[Tex \LARGE \left(x_{1}, y_{1}\right)] is the first point (130,1.25) and *[Tex \LARGE \left(x_{2}, y_{2}\right)] is the second point (140,4)

{{{m=(y[2]-y[1])/(x[2]-x[1])}}} Start with the slope formula.



{{{m=(4-1.25)/(140-130)}}} Plug in {{{y[2]=4}}}, {{{y[1]=1.25}}}, {{{x[2]=140}}}, and {{{x[1]=130}}}



{{{m=(2.75)/(140-130)}}} Subtract {{{1.25}}} from {{{4}}} to get {{{2.75}}}



{{{m=(2.75)/(10)}}} Subtract {{{130}}} from {{{140}}} to get {{{10}}}



{{{m=0.275}}} Divide


So the slope of the line that goes through the points (130,1.25) and (140,4) is {{{m=0.275}}} 



Now let's use the point slope formula:



{{{y-y[1]=m(x-x[1])}}} Start with the point slope formula



{{{y-1.25=0.275(x-130)}}} Plug in {{{m=0.275}}}, {{{x[1]=130}}}, and {{{y[1]=1.25}}}



{{{y-1.25=0.275x+0.275(-130)}}} Distribute



{{{y-1.25=0.275x-35.75}}} Multiply



{{{y=0.275x-35.75+1.25}}} Add {{{1.25}}} to both sides. 



{{{y=0.275x-34.5}}} Combine like terms.



So the equation that goes through the points (130,1.25) and (140,4) is {{{y=0.275x-34.5}}}



The equation is now in slope intercept form {{{y=mx+b}}} where the slope is {{{m=0.275}}} and the y-intercept is {{{b=-34.5}}}




Note:

Since the slope is {{{m=0.275}}}, this means that for every dollar increase that the price per barrel of oil experiences, the price per gallon of gas will increase $0.275

Also, the y-intercept is the value when the price of oil is $0 per barrel. So if the price per barrel is $0, then the price of gas is -34.50 dollars. 







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"Assuming that the relationship is linear, I have to calculate the price for a gallon of gasoline when the oil price reaches $39.34 per barrel"



In this case, we want to know "y" when "x" is equal to 39.34. So simply plug in {{{x=39.34}}} to find "y"


{{{y=0.275x-34.5}}} Start with the equation we just found



{{{y=0.275(39.34)-34.5}}} Plug in {{{x=39.34}}}



{{{y=10.8185-34.5}}} Multiply



{{{y=-23.6815}}} Subtract



So when the price of oil is $39.34 a barrel, the price of gas will be about -23.68 dollars a barrel.



Note: the relationship between the price of oil and the price of gas is a little more complex than just a simple linear relationship (since there are more factors involved).