SOLUTION: Two numbers have a sum of 39. Their difference is 11. What are the two numbers? Yes this is third grade math.

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Question 108858: Two numbers have a sum of 39. Their difference is 11. What are the two numbers? Yes this is third grade math.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
The expression "Two numbers have a sum of 39" is written as

x%2By=39

and "Their difference is 11" is written as

x-y=11

So we get the system


x%2By=39
x-y=11

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=39
1%2Ax-1%2Ay=11

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=39-1%2AxSubtract 1%2Ax from both sides

y=%2839-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=39-1%2Ax Now we've fully isolated y

Since y equals 39-1%2Ax we can substitute the expression 39-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-1%2Ahighlight%28%2839-1%2Ax%29%29=11 Replace y with 39-1%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-1%2A%2839%29-1%28-1%29x=11 Distribute -1 to 39-1%2Ax

1%2Ax-39%2B1%2Ax=11 Multiply



1%2Ax-39%2B1%2Ax=11 Reduce any fractions

1%2Ax%2B1%2Ax=11%2B39Add 39 to both sides


1%2Ax%2B1%2Ax=50 Combine the terms on the right side



2%2Ax=50 Now combine the terms on the left side.


cross%28%281%2F2%29%282%2F1%29%29x=%2850%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2%2F1 and isolate x

So when we multiply 50%2F1 and 1%2F2 (and simplify) we get



x=25 <---------------------------------One answer

Now that we know that x=25, lets substitute that in for x to solve for y

1%2825%29-1%2Ay=11 Plug in x=25 into the 2nd equation

25-1%2Ay=11 Multiply

-1%2Ay=11-25Subtract 25 from both sides

-1%2Ay=-14 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ay=%28-14%2F1%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.

y=-14%2F-1 Multiply the terms on the right side


y=14 Reduce


So this is the other answer


y=14<---------------------------------Other answer


So our solution is

x=25 and y=14

which can also look like

(25,14)

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

1%2Ax%2B1%2Ay=39
1%2Ax-1%2Ay=11

we get


graph of 1%2Ax%2B1%2Ay=39 (red) and 1%2Ax-1%2Ay=11 (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 (25,14). This verifies our answer.


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Check:

Plug in (25,14) into the system of equations


Let x=25 and y=14. Now plug those values into the equation 1%2Ax%2B1%2Ay=39

1%2A%2825%29%2B1%2A%2814%29=39 Plug in x=25 and y=14


25%2B14=39 Multiply


39=39 Add


39=39 Reduce. Since this equation is true the solution works.


So the solution (25,14) satisfies 1%2Ax%2B1%2Ay=39



Let x=25 and y=14. Now plug those values into the equation 1%2Ax-1%2Ay=11

1%2A%2825%29-1%2A%2814%29=11 Plug in x=25 and y=14


25-14=11 Multiply


11=11 Add


11=11 Reduce. Since this equation is true the solution works.


So the solution (25,14) satisfies 1%2Ax-1%2Ay=11


Since the solution (25,14) satisfies the system of equations


1%2Ax%2B1%2Ay=39
1%2Ax-1%2Ay=11


this verifies our answer.