SOLUTION: The sum of two numbers is 53/24 and the difference of the same two numbers is 13/24. What are the numbers? I got lost with the fractions...

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Question 120383: The sum of two numbers is 53/24 and the difference of the same two numbers is 13/24. What are the numbers?
I got lost with the fractions...
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If the "sum of two numbers is 53/24" then we have the first equation x%2By=53%2F24


If the "difference of the same two numbers is 13/24" then we have the second equation x-y=13%2F24


Solved by pluggable solver: Solving a linear system of equations by subsitution


%281%29%2Ax%2B%281%29%2Ay=53%2F24 Start with the first equation


24%28%281%29%2Ax%2B%281%29%2Ay%29=%2824%29%2A%2853%2F24%29 Multiply both sides by the LCD 24



24%2Ax%2B24%2Ay=53Distribute and simplify


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%281%29%2Ax%2B%28-1%29%2Ay=13%2F24 Start with the second equation


24%28%281%29%2Ax%2B%28-1%29%2Ay%29=%2824%29%2A%2813%2F24%29 Multiply both sides by the LCD 24



24%2Ax%2B-24%2Ay=13 Distribute and simplify


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Lets start with the given system of linear equations

24%2Ax%2B24%2Ay=53
24%2Ax-24%2Ay=13

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

24%2Ay=53-24%2AxSubtract 24%2Ax from both sides

y=%2853-24%2Ax%29%2F24 Divide both sides by 24.


Which breaks down and reduces to



y=53%2F24-1%2Ax Now we've fully isolated y

Since y equals 53%2F24-1%2Ax we can substitute the expression 53%2F24-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


24%2Ax%2B-24%2Ahighlight%28%2853%2F24-1%2Ax%29%29=13 Replace y with 53%2F24-1%2Ax. Since this eliminates y, we can now solve for x.

24%2Ax-24%2A%2853%2F24%29-24%28-1%29x=13 Distribute -24 to 53%2F24-1%2Ax

24%2Ax-1272%2F24%2B24%2Ax=13 Multiply



24%2Ax-53%2B24%2Ax=13 Reduce any fractions

24%2Ax%2B24%2Ax=13%2B53Add 53 to both sides


24%2Ax%2B24%2Ax=66 Combine the terms on the right side



48%2Ax=66 Now combine the terms on the left side.


cross%28%281%2F48%29%2848%2F1%29%29x=%2866%2F1%29%281%2F48%29 Multiply both sides by 1%2F48. This will cancel out 48%2F1 and isolate x

So when we multiply 66%2F1 and 1%2F48 (and simplify) we get



x=11%2F8 <---------------------------------One answer

Now that we know that x=11%2F8, lets substitute that in for x to solve for y

24%2811%2F8%29-24%2Ay=13 Plug in x=11%2F8 into the 2nd equation

33-24%2Ay=13 Multiply

-24%2Ay=13-33Subtract 33 from both sides

-24%2Ay=-20 Combine the terms on the right side

cross%28%281%2F-24%29%28-24%29%29%2Ay=%28-20%2F1%29%281%2F-24%29 Multiply both sides by 1%2F-24. This will cancel out -24 on the left side.

y=-20%2F-24 Multiply the terms on the right side


y=5%2F6 Reduce


So this is the other answer


y=5%2F6<---------------------------------Other answer


So our solution is

x=11%2F8 and y=5%2F6

which can also look like

(11%2F8,5%2F6)

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

24%2Ax%2B24%2Ay=53
24%2Ax-24%2Ay=13

we get


graph of 24%2Ax%2B24%2Ay=53 (red) and 24%2Ax-24%2Ay=13 (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 (11%2F8,5%2F6). This verifies our answer.


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Check:

Plug in (11%2F8,5%2F6) into the system of equations


Let x=11%2F8 and y=5%2F6. Now plug those values into the equation 24%2Ax%2B24%2Ay=53

24%2A%2811%2F8%29%2B24%2A%285%2F6%29=53 Plug in x=11%2F8 and y=5%2F6


264%2F8%2B120%2F6=53 Multiply


1272%2F24=53 Add


53=53 Reduce. Since this equation is true the solution works.


So the solution (11%2F8,5%2F6) satisfies 24%2Ax%2B24%2Ay=53



Let x=11%2F8 and y=5%2F6. Now plug those values into the equation 24%2Ax-24%2Ay=13

24%2A%2811%2F8%29-24%2A%285%2F6%29=13 Plug in x=11%2F8 and y=5%2F6


264%2F8-120%2F6=13 Multiply


312%2F24=13 Add


13=13 Reduce. Since this equation is true the solution works.


So the solution (11%2F8,5%2F6) satisfies 24%2Ax-24%2Ay=13


Since the solution (11%2F8,5%2F6) satisfies the system of equations


24%2Ax%2B24%2Ay=53
24%2Ax-24%2Ay=13


this verifies our answer.





The two numbers are 11%2F8 and 5%2F6