SOLUTION: A man drives out in the country at an average rate of 40 mph. In the evening traffic he drives back at 30 mph. If the total traveling time is 7 hours, what is the distance traveled

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Question 904478: A man drives out in the country at an average rate of 40 mph. In the evening traffic he drives back at 30 mph. If the total traveling time is 7 hours, what is the distance traveled?
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
a+b=7
40a-30b=0
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=7
40%2Ax-30%2Ay=0

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 40 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 40 is 40, we need to multiply both sides of the top equation by 40 and multiply both sides of the bottom equation by -1 like this:

40%2A%281%2Ax%2B1%2Ay%29=%287%29%2A40 Multiply the top equation (both sides) by 40
-1%2A%2840%2Ax-30%2Ay%29=%280%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
40%2Ax%2B40%2Ay=280
-40%2Ax%2B30%2Ay=0

Notice how 40 and -40 add to zero (ie 40%2B-40=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2840%2Ax-40%2Ax%29%2B%2840%2Ay%2B30%2Ay%29=280%2B0

%2840-40%29%2Ax%2B%2840%2B30%29y=280%2B0

cross%2840%2B-40%29%2Ax%2B%2840%2B30%29%2Ay=280%2B0 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

70%2Ay=280

y=280%2F70 Divide both sides by 70 to solve for y



y=4 Reduce


Now plug this answer into the top equation 1%2Ax%2B1%2Ay=7 to solve for x

1%2Ax%2B1%284%29=7 Plug in y=4


1%2Ax%2B4=7 Multiply



1%2Ax=7-4 Subtract 4 from both sides

1%2Ax=3 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%283%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=3 Multiply the terms on the right side


So our answer is

x=3, y=4

which also looks like

(3, 4)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax%2B1%2Ay=7
40%2Ax-30%2Ay=0

we get



graph of 1%2Ax%2B1%2Ay=7 (red) 40%2Ax-30%2Ay=0 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (3,4). This verifies our answer.