SOLUTION: x+2y=54 x+5y=110 Find x and y Haven't done this for 45 years, and have forgotten how! Many thanks

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Question 705217: x+2y=54
x+5y=110
Find x and y
Haven't done this for 45 years, and have forgotten how!
Many thanks
Answer by mouk(232) 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%2B2%2Ay=54
1%2Ax%2B5%2Ay=110

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

2%2Ay=54-1%2AxSubtract 1%2Ax from both sides

y=%2854-1%2Ax%29%2F2 Divide both sides by 2.


Which breaks down and reduces to



y=27-%281%2F2%29%2Ax Now we've fully isolated y

Since y equals 27-%281%2F2%29%2Ax we can substitute the expression 27-%281%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B5%2Ahighlight%28%2827-%281%2F2%29%2Ax%29%29=110 Replace y with 27-%281%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B5%2A%2827%29%2B5%28-1%2F2%29x=110 Distribute 5 to 27-%281%2F2%29%2Ax

1%2Ax%2B135-%285%2F2%29%2Ax=110 Multiply



1%2Ax%2B135-%285%2F2%29%2Ax=110 Reduce any fractions

1%2Ax-%285%2F2%29%2Ax=110-135 Subtract 135 from both sides


1%2Ax-%285%2F2%29%2Ax=-25 Combine the terms on the right side



%282%2F2%29%2Ax-%285%2F2%29x=-25 Make 1 into a fraction with a denominator of 2

%28-3%2F2%29%2Ax=-25 Now combine the terms on the left side.


cross%28%282%2F-3%29%28-3%2F2%29%29x=%28-25%2F1%29%282%2F-3%29 Multiply both sides by 2%2F-3. This will cancel out -3%2F2 and isolate x

So when we multiply -25%2F1 and 2%2F-3 (and simplify) we get



x=50%2F3 <---------------------------------One answer

Now that we know that x=50%2F3, lets substitute that in for x to solve for y

1%2850%2F3%29%2B5%2Ay=110 Plug in x=50%2F3 into the 2nd equation

50%2F3%2B5%2Ay=110 Multiply

5%2Ay=110-50%2F3Subtract 50%2F3 from both sides

5%2Ay=330%2F3-50%2F3 Make 110 into a fraction with a denominator of 3



5%2Ay=280%2F3 Combine the terms on the right side

cross%28%281%2F5%29%285%29%29%2Ay=%28280%2F3%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5 on the left side.

y=280%2F15 Multiply the terms on the right side


y=56%2F3 Reduce


So this is the other answer


y=56%2F3<---------------------------------Other answer


So our solution is

x=50%2F3 and y=56%2F3

which can also look like

(50%2F3,56%2F3)

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

1%2Ax%2B2%2Ay=54
1%2Ax%2B5%2Ay=110

we get


graph of 1%2Ax%2B2%2Ay=54 (red) and 1%2Ax%2B5%2Ay=110 (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 (50%2F3,56%2F3). This verifies our answer.


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Check:

Plug in (50%2F3,56%2F3) into the system of equations


Let x=50%2F3 and y=56%2F3. Now plug those values into the equation 1%2Ax%2B2%2Ay=54

1%2A%2850%2F3%29%2B2%2A%2856%2F3%29=54 Plug in x=50%2F3 and y=56%2F3


50%2F3%2B112%2F3=54 Multiply


162%2F3=54 Add


54=54 Reduce. Since this equation is true the solution works.


So the solution (50%2F3,56%2F3) satisfies 1%2Ax%2B2%2Ay=54



Let x=50%2F3 and y=56%2F3. Now plug those values into the equation 1%2Ax%2B5%2Ay=110

1%2A%2850%2F3%29%2B5%2A%2856%2F3%29=110 Plug in x=50%2F3 and y=56%2F3


50%2F3%2B280%2F3=110 Multiply


330%2F3=110 Add


110=110 Reduce. Since this equation is true the solution works.


So the solution (50%2F3,56%2F3) satisfies 1%2Ax%2B5%2Ay=110


Since the solution (50%2F3,56%2F3) satisfies the system of equations


1%2Ax%2B2%2Ay=54
1%2Ax%2B5%2Ay=110


this verifies our answer.