SOLUTION: Please Help. Solve the following problem. Number problem. The sum of two numbers is 40. If their difference is 10, find the two numbers. Please show the equation used. Th

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Question 114448: Please Help.
Solve the following problem.
Number problem.
The sum of two numbers is 40. If their difference is 10, find the two numbers.
Please show the equation used.
Thanks
JMS
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If the sum of two numbers is 40, then we have the equation x%2By=40 and their difference is 10 then we get another equation x-y=10



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=40
1%2Ax-1%2Ay=10

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=40-1%2AxSubtract 1%2Ax from both sides

y=%2840-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=40-1%2Ax Now we've fully isolated y

Since y equals 40-1%2Ax we can substitute the expression 40-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-1%2Ahighlight%28%2840-1%2Ax%29%29=10 Replace y with 40-1%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-1%2A%2840%29-1%28-1%29x=10 Distribute -1 to 40-1%2Ax

1%2Ax-40%2B1%2Ax=10 Multiply



1%2Ax-40%2B1%2Ax=10 Reduce any fractions

1%2Ax%2B1%2Ax=10%2B40Add 40 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=10 Plug in x=25 into the 2nd equation

25-1%2Ay=10 Multiply

-1%2Ay=10-25Subtract 25 from both sides

-1%2Ay=-15 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ay=%28-15%2F1%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.

y=-15%2F-1 Multiply the terms on the right side


y=15 Reduce


So this is the other answer


y=15<---------------------------------Other answer


So our solution is

x=25 and y=15

which can also look like

(25,15)

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

1%2Ax%2B1%2Ay=40
1%2Ax-1%2Ay=10

we get


graph of 1%2Ax%2B1%2Ay=40 (red) and 1%2Ax-1%2Ay=10 (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 (25,15). This verifies our answer.


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Check:

Plug in (25,15) into the system of equations


Let x=25 and y=15. Now plug those values into the equation 1%2Ax%2B1%2Ay=40

1%2A%2825%29%2B1%2A%2815%29=40 Plug in x=25 and y=15


25%2B15=40 Multiply


40=40 Add


40=40 Reduce. Since this equation is true the solution works.


So the solution (25,15) satisfies 1%2Ax%2B1%2Ay=40



Let x=25 and y=15. Now plug those values into the equation 1%2Ax-1%2Ay=10

1%2A%2825%29-1%2A%2815%29=10 Plug in x=25 and y=15


25-15=10 Multiply


10=10 Add


10=10 Reduce. Since this equation is true the solution works.


So the solution (25,15) satisfies 1%2Ax-1%2Ay=10


Since the solution (25,15) satisfies the system of equations


1%2Ax%2B1%2Ay=40
1%2Ax-1%2Ay=10


this verifies our answer.