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) About Me  (Show Source):
You can put this solution on YOUR website!
Translate the word problem into a system of equations

x%2By=27 "The sum of two numbers is 27"
%281%2F2%29x%2B%281%2F3%29y=11 "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


%281%2F2%29%2Ax%2B%281%2F3%29%2Ay=11 Start with the second equation


6%28%281%2F2%29%2Ax%2B%281%2F3%29%2Ay%29=%286%29%2A%2811%29 Multiply both sides by the LCD 6



3%2Ax%2B2%2Ay=66 Distribute and simplify


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Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=27
3%2Ax%2B2%2Ay=66

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=27-1%2AxSubtract 1%2Ax from both sides

y=%2827-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=27-1%2Ax Now we've fully isolated y

Since y equals 27-1%2Ax we can substitute the expression 27-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B2%2Ahighlight%28%2827-1%2Ax%29%29=66 Replace y with 27-1%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax%2B2%2A%2827%29%2B2%28-1%29x=66 Distribute 2 to 27-1%2Ax

3%2Ax%2B54-2%2Ax=66 Multiply



3%2Ax%2B54-2%2Ax=66 Reduce any fractions

3%2Ax-2%2Ax=66-54 Subtract 54 from both sides


3%2Ax-2%2Ax=12 Combine the terms on the right side



1%2Ax=12 Now combine the terms on the left side.


cross%28%281%2F1%29%281%2F1%29%29x=%2812%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1%2F1 and isolate x

So when we multiply 12%2F1 and 1%2F1 (and simplify) we get



x=12 <---------------------------------One answer

Now that we know that x=12, lets substitute that in for x to solve for y

3%2812%29%2B2%2Ay=66 Plug in x=12 into the 2nd equation

36%2B2%2Ay=66 Multiply

2%2Ay=66-36Subtract 36 from both sides

2%2Ay=30 Combine the terms on the right side

cross%28%281%2F2%29%282%29%29%2Ay=%2830%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.

y=30%2F2 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=12 and y=15

which can also look like

(12,15)

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

1%2Ax%2B1%2Ay=27
3%2Ax%2B2%2Ay=66

we get


graph of 1%2Ax%2B1%2Ay=27 (red) and 3%2Ax%2B2%2Ay=66 (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 (12,15). This verifies our answer.


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Check:

Plug in (12,15) into the system of equations


Let x=12 and y=15. Now plug those values into the equation 1%2Ax%2B1%2Ay=27

1%2A%2812%29%2B1%2A%2815%29=27 Plug in x=12 and y=15


12%2B15=27 Multiply


27=27 Add


27=27 Reduce. Since this equation is true the solution works.


So the solution (12,15) satisfies 1%2Ax%2B1%2Ay=27



Let x=12 and y=15. Now plug those values into the equation 3%2Ax%2B2%2Ay=66

3%2A%2812%29%2B2%2A%2815%29=66 Plug in x=12 and y=15


36%2B30=66 Multiply


66=66 Add


66=66 Reduce. Since this equation is true the solution works.


So the solution (12,15) satisfies 3%2Ax%2B2%2Ay=66


Since the solution (12,15) satisfies the system of equations


1%2Ax%2B1%2Ay=27
3%2Ax%2B2%2Ay=66


this verifies our answer.





So our two numbers are: 12 and 15

Notice 12+15=27 and %281%2F2%2912%2B%281%2F3%2915=6%2B5=11. So this verifies our answer.