SOLUTION: Your math teacher tells you that next week's test is worth 100 points and contains 38 problems. Each problem is worth either 5 points or 2 points. Because you are studying for syst

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Question 123441: Your math teacher tells you that next week's test is worth 100 points and contains 38 problems. Each problem is worth either 5 points or 2 points. Because you are studying for systems of linear equations, your teacher says that for extra credit you can figure out how many problems of each value are on the test. How many of each value are there?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let x= # of five-point problems, y= # of two-point problems

Since there are 38 problems, this means we have the first equation x%2By=38

Also, since the total sum of the five-point and two-point problems is 100, this tells us that the second equation is 5x%2B2y=100


So we have the system

x%2By=38
5x%2B2y=100




Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=38
5%2Ax%2B2%2Ay=100

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=38-1%2AxSubtract 1%2Ax from both sides

y=%2838-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=38-1%2Ax Now we've fully isolated y

Since y equals 38-1%2Ax we can substitute the expression 38-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


5%2Ax%2B2%2Ahighlight%28%2838-1%2Ax%29%29=100 Replace y with 38-1%2Ax. Since this eliminates y, we can now solve for x.

5%2Ax%2B2%2A%2838%29%2B2%28-1%29x=100 Distribute 2 to 38-1%2Ax

5%2Ax%2B76-2%2Ax=100 Multiply



5%2Ax%2B76-2%2Ax=100 Reduce any fractions

5%2Ax-2%2Ax=100-76 Subtract 76 from both sides


5%2Ax-2%2Ax=24 Combine the terms on the right side



3%2Ax=24 Now combine the terms on the left side.


cross%28%281%2F3%29%283%2F1%29%29x=%2824%2F1%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3%2F1 and isolate x

So when we multiply 24%2F1 and 1%2F3 (and simplify) we get



x=8 <---------------------------------One answer

Now that we know that x=8, lets substitute that in for x to solve for y

5%288%29%2B2%2Ay=100 Plug in x=8 into the 2nd equation

40%2B2%2Ay=100 Multiply

2%2Ay=100-40Subtract 40 from both sides

2%2Ay=60 Combine the terms on the right side

cross%28%281%2F2%29%282%29%29%2Ay=%2860%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.

y=60%2F2 Multiply the terms on the right side


y=30 Reduce


So this is the other answer


y=30<---------------------------------Other answer


So our solution is

x=8 and y=30

which can also look like

(8,30)

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

1%2Ax%2B1%2Ay=38
5%2Ax%2B2%2Ay=100

we get


graph of 1%2Ax%2B1%2Ay=38 (red) and 5%2Ax%2B2%2Ay=100 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (8,30). This verifies our answer.


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Check:

Plug in (8,30) into the system of equations


Let x=8 and y=30. Now plug those values into the equation 1%2Ax%2B1%2Ay=38

1%2A%288%29%2B1%2A%2830%29=38 Plug in x=8 and y=30


8%2B30=38 Multiply


38=38 Add


38=38 Reduce. Since this equation is true the solution works.


So the solution (8,30) satisfies 1%2Ax%2B1%2Ay=38



Let x=8 and y=30. Now plug those values into the equation 5%2Ax%2B2%2Ay=100

5%2A%288%29%2B2%2A%2830%29=100 Plug in x=8 and y=30


40%2B60=100 Multiply


100=100 Add


100=100 Reduce. Since this equation is true the solution works.


So the solution (8,30) satisfies 5%2Ax%2B2%2Ay=100


Since the solution (8,30) satisfies the system of equations


1%2Ax%2B1%2Ay=38
5%2Ax%2B2%2Ay=100


this verifies our answer.






So there are 8 five-point problems and 30 two-point problems