Expert: Steve Holleran - 3/20/2008

Question
2.) Find the values of x and y that solve the following system of equations:We call the two numbers x & y.
7x + 9y = 9
6x + 5y = -14
x =
y =

4.) Find the values of x and y that solve the following system of equations:We call the two numbers x and y.
7x - 6y = 5
-2x + 5y = -8
x =
y =

5.) Find two numbers whose sum is 44 and such that one is 3 times as large as the other. We call the two numbers x and y.
x =
y =

6.) In a family there are two cars. The sum of the average miles per gallon obtained by the two cars in a particular week is 50. The first car has consumed 40 gallons during that week, and the second has consumed 30 gallons, for a total of 1800 miles driven by the two cars combined. What was the average gas mileage obtained by each of the two cars in that week?
First Car: ___miles/gallon
Second Car: ___miles/gallon

Can you please help me with the above problems. The teachers states the way I'm setting up and answering my solution is all wrong. Thank you!


Answer
Hi Elisha,

Basically what you want to do here is to use the process called "elimination".  We need to use multiplication to get the x or y coefficients to be opposites:

In #2, let's eliminate the x terms.  To do this, let's multiply the top equation by -6 and the bottom one by 7:

7x + 9y = 9 ----(-6)--->  -42x - 54y = -54

6x + 5y = -14---(7)---->   42x + 35y = -98

Now add:                         -19y = -152

so                                  y = -152/-19 = 8

then use one of the original equations to find x:

7x + 9y = 9  -----> 7x + 9(8) = 9

                    7x + 72 = 9

                    7x = -63     so x = -9.



Do the same for #4.  We can multiply the top one by 2 and the bottom one by 7:

7x - 6y = 5 -------(2)--------> 14x - 12y = 10

-2x + 5y = -8 -----(7)-------->-14x + 35y = -56

Add:                                  23y = -46  so y = -2

Then  7x - 6(-2) = 5

     7x + 12 = 5    so 7x = -7  and x = -1


For #5, if the sum is 44, you have x + y = 44

One is 3 times the other gives     x = 3y.

Here, its easier just to substitute 3y into the first equation for x:

            x + y = 44 --------> 3y + y = 44

                                 4y = 44  so y = 11, then x = 33


For #6, Let x = # miles driven in car 1

           y = # miles driven in car 2.

Then you have :             x + y = 1800

Then for the average miles per gallon, since x used 40 gallons, its
average MPG is x/40.  The second car's is y/30.  So

               x/40 + y/30 = 50

Now here, I would multiply through the equation by 120, the common denominator, to get rid of the fractions:

            120 * x/40 + 120 * y/30 = 120 * 50

so               3x + 4y = 6000

Now take the other equation and multiply it by -3:

  x + y = 1800 -------(-3)-------> -3x -3y = -5400

add                                  3x + 4y = 6000

                                          y = 600

so x = 1800 - 600 = 1200

And the average MPG for each car is :  Car 1: 1200/40 = 30 mpg

                                      Car 2:  600/30 = 20 mpg.


Hope this helps.

Steve