SOLUTION: The sum of two numbers is 111, and their difference is 63. What are the two numbers? *what is the formula?

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Question 109400This question is from textbook
: The sum of two numbers is 111, and their difference is 63.
What are the two numbers?
*what is the formula? This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of two numbers can be written as x%2By=111 and the difference can be written as x-y=63 which makes the system

x%2By=111
x-y=63


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=111
1%2Ax-1%2Ay=63

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=111-1%2AxSubtract 1%2Ax from both sides

y=%28111-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=111-1%2Ax Now we've fully isolated y

Since y equals 111-1%2Ax we can substitute the expression 111-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-1%2Ahighlight%28%28111-1%2Ax%29%29=63 Replace y with 111-1%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-1%2A%28111%29-1%28-1%29x=63 Distribute -1 to 111-1%2Ax

1%2Ax-111%2B1%2Ax=63 Multiply



1%2Ax-111%2B1%2Ax=63 Reduce any fractions

1%2Ax%2B1%2Ax=63%2B111Add 111 to both sides


1%2Ax%2B1%2Ax=174 Combine the terms on the right side



2%2Ax=174 Now combine the terms on the left side.


cross%28%281%2F2%29%282%2F1%29%29x=%28174%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2%2F1 and isolate x

So when we multiply 174%2F1 and 1%2F2 (and simplify) we get



x=87 <---------------------------------One answer

Now that we know that x=87, lets substitute that in for x to solve for y

1%2887%29-1%2Ay=63 Plug in x=87 into the 2nd equation

87-1%2Ay=63 Multiply

-1%2Ay=63-87Subtract 87 from both sides

-1%2Ay=-24 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ay=%28-24%2F1%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.

y=-24%2F-1 Multiply the terms on the right side


y=24 Reduce


So this is the other answer


y=24<---------------------------------Other answer


So our solution is

x=87 and y=24

which can also look like

(87,24)

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

1%2Ax%2B1%2Ay=111
1%2Ax-1%2Ay=63

we get


graph of 1%2Ax%2B1%2Ay=111 (red) and 1%2Ax-1%2Ay=63 (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 (87,24). This verifies our answer.


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Check:

Plug in (87,24) into the system of equations


Let x=87 and y=24. Now plug those values into the equation 1%2Ax%2B1%2Ay=111

1%2A%2887%29%2B1%2A%2824%29=111 Plug in x=87 and y=24


87%2B24=111 Multiply


111=111 Add


111=111 Reduce. Since this equation is true the solution works.


So the solution (87,24) satisfies 1%2Ax%2B1%2Ay=111



Let x=87 and y=24. Now plug those values into the equation 1%2Ax-1%2Ay=63

1%2A%2887%29-1%2A%2824%29=63 Plug in x=87 and y=24


87-24=63 Multiply


63=63 Add


63=63 Reduce. Since this equation is true the solution works.


So the solution (87,24) satisfies 1%2Ax-1%2Ay=63


Since the solution (87,24) satisfies the system of equations


1%2Ax%2B1%2Ay=111
1%2Ax-1%2Ay=63


this verifies our answer.





So our two numbers are 87 and 24