SOLUTION: find two numbers of which the sum is 11 and the difference is 5.

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Question 88243: find two numbers of which the sum is 11 and the difference is 5.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
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=11
1%2Ax-1%2Ay=5

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=11-1%2AxSubtract 1%2Ax from both sides

y=%2811-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=11-1%2Ax Now we've fully isolated y

Since y equals 11-1%2Ax we can substitute the expression 11-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-1%2Ahighlight%28%2811-1%2Ax%29%29=5 Replace y with 11-1%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-1%2A%2811%29-1%28-1%29x=5 Distribute -1 to 11-1%2Ax

1%2Ax-11%2B1%2Ax=5 Multiply



1%2Ax-11%2B1%2Ax=5 Reduce any fractions

1%2Ax%2B1%2Ax=5%2B11Add 11 to both sides


1%2Ax%2B1%2Ax=16 Combine the terms on the right side



2%2Ax=16 Now combine the terms on the left side.


cross%28%281%2F2%29%282%2F1%29%29x=%2816%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2%2F1 and isolate x

So when we multiply 16%2F1 and 1%2F2 (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

1%288%29-1%2Ay=5 Plug in x=8 into the 2nd equation

8-1%2Ay=5 Multiply

-1%2Ay=5-8Subtract 8 from both sides

-1%2Ay=-3 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ay=%28-3%2F1%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.

y=-3%2F-1 Multiply the terms on the right side


y=3 Reduce


So this is the other answer


y=3<---------------------------------Other answer


So our solution is

x=8 and y=3

which can also look like

(8,3)

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

1%2Ax%2B1%2Ay=11
1%2Ax-1%2Ay=5

we get


graph of 1%2Ax%2B1%2Ay=11 (red) and 1%2Ax-1%2Ay=5 (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 (8,3). This verifies our answer.


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Check:

Plug in (8,3) into the system of equations


Let x=8 and y=3. Now plug those values into the equation 1%2Ax%2B1%2Ay=11

1%2A%288%29%2B1%2A%283%29=11 Plug in x=8 and y=3


8%2B3=11 Multiply


11=11 Add


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


So the solution (8,3) satisfies 1%2Ax%2B1%2Ay=11



Let x=8 and y=3. Now plug those values into the equation 1%2Ax-1%2Ay=5

1%2A%288%29-1%2A%283%29=5 Plug in x=8 and y=3


8-3=5 Multiply


5=5 Add


5=5 Reduce. Since this equation is true the solution works.


So the solution (8,3) satisfies 1%2Ax-1%2Ay=5


Since the solution (8,3) satisfies the system of equations


1%2Ax%2B1%2Ay=11
1%2Ax-1%2Ay=5


this verifies our answer.





So the 2 numbers are 8 and 3