SOLUTION: The sum of two numbers is 42. Their difference is 6. What are the two numbers?

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Question 111438: The sum of two numbers is 42. Their difference is 6. What are the two numbers?
Found 2 solutions by jim_thompson5910, edjones:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Translate the given info:
"The sum of two numbers is 42"-->x%2By=42



"Their difference is 6"-->x-y=6


So we get the system
x%2By=42
x-y=6


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=42
1%2Ax-1%2Ay=6

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=42-1%2AxSubtract 1%2Ax from both sides

y=%2842-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=42-1%2Ax Now we've fully isolated y

Since y equals 42-1%2Ax we can substitute the expression 42-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-1%2Ahighlight%28%2842-1%2Ax%29%29=6 Replace y with 42-1%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-1%2A%2842%29-1%28-1%29x=6 Distribute -1 to 42-1%2Ax

1%2Ax-42%2B1%2Ax=6 Multiply



1%2Ax-42%2B1%2Ax=6 Reduce any fractions

1%2Ax%2B1%2Ax=6%2B42Add 42 to both sides


1%2Ax%2B1%2Ax=48 Combine the terms on the right side



2%2Ax=48 Now combine the terms on the left side.


cross%28%281%2F2%29%282%2F1%29%29x=%2848%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2%2F1 and isolate x

So when we multiply 48%2F1 and 1%2F2 (and simplify) we get



x=24 <---------------------------------One answer

Now that we know that x=24, lets substitute that in for x to solve for y

1%2824%29-1%2Ay=6 Plug in x=24 into the 2nd equation

24-1%2Ay=6 Multiply

-1%2Ay=6-24Subtract 24 from both sides

-1%2Ay=-18 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ay=%28-18%2F1%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.

y=-18%2F-1 Multiply the terms on the right side


y=18 Reduce


So this is the other answer


y=18<---------------------------------Other answer


So our solution is

x=24 and y=18

which can also look like

(24,18)

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

1%2Ax%2B1%2Ay=42
1%2Ax-1%2Ay=6

we get


graph of 1%2Ax%2B1%2Ay=42 (red) and 1%2Ax-1%2Ay=6 (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 (24,18). This verifies our answer.


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

Plug in (24,18) into the system of equations


Let x=24 and y=18. Now plug those values into the equation 1%2Ax%2B1%2Ay=42

1%2A%2824%29%2B1%2A%2818%29=42 Plug in x=24 and y=18


24%2B18=42 Multiply


42=42 Add


42=42 Reduce. Since this equation is true the solution works.


So the solution (24,18) satisfies 1%2Ax%2B1%2Ay=42



Let x=24 and y=18. Now plug those values into the equation 1%2Ax-1%2Ay=6

1%2A%2824%29-1%2A%2818%29=6 Plug in x=24 and y=18


24-18=6 Multiply


6=6 Add


6=6 Reduce. Since this equation is true the solution works.


So the solution (24,18) satisfies 1%2Ax-1%2Ay=6


Since the solution (24,18) satisfies the system of equations


1%2Ax%2B1%2Ay=42
1%2Ax-1%2Ay=6


this verifies our answer.






So our two numbers are 24 and 18

Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
A) x+y=42
B) x-y=6
.
B) x-y+y=6+y
x=y+6
A)y+6+y=42 substitute y+6 for x.
2y+6-6=42-6
2y=36
2y/2=36/2
y=18
A) x+18=42
x+18-18=42-18
x=24
Check:
B) 24-18=6 true
.
Ed