SOLUTION: The sum of two numbers is 50 and their difference is 18. Find the numbers. Im so confused could someone please help?

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Question 120563: The sum of two numbers is 50 and their difference is 18. Find the numbers.
Im so confused could someone please help?
Found 2 solutions by jim_thompson5910, solver91311:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If the "sum of two numbers is 50", then we have the first equation x%2By=50



If "their difference is 18", then we have the second equation x-y=18



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=50
1%2Ax-1%2Ay=18

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=50-1%2AxSubtract 1%2Ax from both sides

y=%2850-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=50-1%2Ax Now we've fully isolated y

Since y equals 50-1%2Ax we can substitute the expression 50-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-1%2Ahighlight%28%2850-1%2Ax%29%29=18 Replace y with 50-1%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-1%2A%2850%29-1%28-1%29x=18 Distribute -1 to 50-1%2Ax

1%2Ax-50%2B1%2Ax=18 Multiply



1%2Ax-50%2B1%2Ax=18 Reduce any fractions

1%2Ax%2B1%2Ax=18%2B50Add 50 to both sides


1%2Ax%2B1%2Ax=68 Combine the terms on the right side



2%2Ax=68 Now combine the terms on the left side.


cross%28%281%2F2%29%282%2F1%29%29x=%2868%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2%2F1 and isolate x

So when we multiply 68%2F1 and 1%2F2 (and simplify) we get



x=34 <---------------------------------One answer

Now that we know that x=34, lets substitute that in for x to solve for y

1%2834%29-1%2Ay=18 Plug in x=34 into the 2nd equation

34-1%2Ay=18 Multiply

-1%2Ay=18-34Subtract 34 from both sides

-1%2Ay=-16 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ay=%28-16%2F1%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.

y=-16%2F-1 Multiply the terms on the right side


y=16 Reduce


So this is the other answer


y=16<---------------------------------Other answer


So our solution is

x=34 and y=16

which can also look like

(34,16)

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

1%2Ax%2B1%2Ay=50
1%2Ax-1%2Ay=18

we get


graph of 1%2Ax%2B1%2Ay=50 (red) and 1%2Ax-1%2Ay=18 (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 (34,16). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (34,16) into the system of equations


Let x=34 and y=16. Now plug those values into the equation 1%2Ax%2B1%2Ay=50

1%2A%2834%29%2B1%2A%2816%29=50 Plug in x=34 and y=16


34%2B16=50 Multiply


50=50 Add


50=50 Reduce. Since this equation is true the solution works.


So the solution (34,16) satisfies 1%2Ax%2B1%2Ay=50



Let x=34 and y=16. Now plug those values into the equation 1%2Ax-1%2Ay=18

1%2A%2834%29-1%2A%2816%29=18 Plug in x=34 and y=16


34-16=18 Multiply


18=18 Add


18=18 Reduce. Since this equation is true the solution works.


So the solution (34,16) satisfies 1%2Ax-1%2Ay=18


Since the solution (34,16) satisfies the system of equations


1%2Ax%2B1%2Ay=50
1%2Ax-1%2Ay=18


this verifies our answer.





So the two numbers are 34 and 16

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
Let's call one of the numbers x and the other one y. Their sum is then x%2By, and their difference is x-y.

The sum of two numbers, x%2By, is, (=), 50, so x%2By=50.

The difference of the same two numbers, x-y, is, (=), 18, so x-y=18

Now you have a system of two equations in two variables. There are a number of methods to solve these systems, but this one lends itself to the Gaussian Elimination method. In this case, just add the two equations term by term which will eliminate the y variable, thus:

x%2By=50
x-y=18
x%2Bx%2By-y=50%2B18
2x%2B0y=68
2x%2F2=68%2F2
x=34
So one of the numbers has to be 34. You can get the other just by subtracting 34 from 50 because of the relationship x%2By=50, but I'm going to show the elimination method on the original two equations to eliminate the x variable this time.

x%2By=50
x-y=18

Step 1: Multiply the bottom equation by -1

x%2By=50
-x%2By=-18

Now add them term by term:

x-x%2By%2By=50-18
0x%2B2y=32
2y%2F2=32%2F2
y=16

Check your answer:

34 + 16 = 50

34 - 16 = 18.

Answer checks.

Hope this helps.
John