SOLUTION: Solve for X and Y in the following problems. a. X + Y=6 ; 2X + Y = 8

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Question 120401: Solve for X and Y in the following problems.
a. X + Y=6 ; 2X + Y = 8

Answer by jim_thompson5910(35256) 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%2B1%2Ay=6
2%2Ax%2B1%2Ay=8

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=6-1%2AxSubtract 1%2Ax from both sides

y=%286-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=6-1%2Ax Now we've fully isolated y

Since y equals 6-1%2Ax we can substitute the expression 6-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B1%2Ahighlight%28%286-1%2Ax%29%29=8 Replace y with 6-1%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax%2B1%2A%286%29%2B1%28-1%29x=8 Distribute 1 to 6-1%2Ax

2%2Ax%2B6-1%2Ax=8 Multiply



2%2Ax%2B6-1%2Ax=8 Reduce any fractions

2%2Ax-1%2Ax=8-6 Subtract 6 from both sides


2%2Ax-1%2Ax=2 Combine the terms on the right side



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


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

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



x=2 <---------------------------------One answer

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

2%282%29%2B1%2Ay=8 Plug in x=2 into the 2nd equation

4%2B1%2Ay=8 Multiply

1%2Ay=8-4Subtract 4 from both sides

1%2Ay=4 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%284%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=4%2F1 Multiply the terms on the right side


y=4 Reduce


So this is the other answer


y=4<---------------------------------Other answer


So our solution is

x=2 and y=4

which can also look like

(2,4)

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

1%2Ax%2B1%2Ay=6
2%2Ax%2B1%2Ay=8

we get


graph of 1%2Ax%2B1%2Ay=6 (red) and 2%2Ax%2B1%2Ay=8 (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 (2,4). This verifies our answer.


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Check:

Plug in (2,4) into the system of equations


Let x=2 and y=4. Now plug those values into the equation 1%2Ax%2B1%2Ay=6

1%2A%282%29%2B1%2A%284%29=6 Plug in x=2 and y=4


2%2B4=6 Multiply


6=6 Add


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


So the solution (2,4) satisfies 1%2Ax%2B1%2Ay=6



Let x=2 and y=4. Now plug those values into the equation 2%2Ax%2B1%2Ay=8

2%2A%282%29%2B1%2A%284%29=8 Plug in x=2 and y=4


4%2B4=8 Multiply


8=8 Add


8=8 Reduce. Since this equation is true the solution works.


So the solution (2,4) satisfies 2%2Ax%2B1%2Ay=8


Since the solution (2,4) satisfies the system of equations


1%2Ax%2B1%2Ay=6
2%2Ax%2B1%2Ay=8


this verifies our answer.