SOLUTION: The sum of two numbers is 27. one half of the first number plus one third of the second number is 11. Find the numbers

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Question 103929: The sum of two numbers is 27. one half of the first number plus one third of the second number is 11. Find the numbers
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Translate the word problem into a system of equations

"The sum of two numbers is 27"
"one half of the first number plus one third of the second number is 11"


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


Start with the second equation


Multiply both sides by the LCD 6



Distribute and simplify


-----------------------------------------



Lets start with the given system of linear equations




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

Subtract from both sides

Divide both sides by 1.


Which breaks down and reduces to



Now we've fully isolated y

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


Replace y with . Since this eliminates y, we can now solve for x.

Distribute 2 to

Multiply



Reduce any fractions

Subtract from both sides


Combine the terms on the right side



Now combine the terms on the left side.


Multiply both sides by . This will cancel out and isolate x

So when we multiply and (and simplify) we get



<---------------------------------One answer

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

Plug in into the 2nd equation

Multiply

Subtract from both sides

Combine the terms on the right side

Multiply both sides by . This will cancel out 2 on the left side.

Multiply the terms on the right side


Reduce


So this is the other answer


<---------------------------------Other answer


So our solution is

and

which can also look like

(,)

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




we get


graph of (red) and (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 (,). This verifies our answer.


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

Plug in (,) into the system of equations


Let and . Now plug those values into the equation

Plug in and


Multiply


Add


Reduce. Since this equation is true the solution works.


So the solution (,) satisfies



Let and . Now plug those values into the equation

Plug in and


Multiply


Add


Reduce. Since this equation is true the solution works.


So the solution (,) satisfies


Since the solution (,) satisfies the system of equations






this verifies our answer.





So our two numbers are: 12 and 15

Notice 12+15=27 and . So this verifies our answer.
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