SOLUTION: -2x+3y=14 x+2y=7

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Question 93526: -2x+3y=14
x+2y=7
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you want to solve the system right?



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


Lets start with the given system of linear equations

-2%2Ax%2B3%2Ay=14
1%2Ax%2B2%2Ay=7

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

3%2Ay=14%2B2%2AxAdd 2%2Ax to both sides

y=%2814%2B2%2Ax%29%2F3 Divide both sides by 3.


Which breaks down and reduces to



y=14%2F3%2B%282%2F3%29%2Ax Now we've fully isolated y

Since y equals 14%2F3%2B%282%2F3%29%2Ax we can substitute the expression 14%2F3%2B%282%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B2%2Ahighlight%28%2814%2F3%2B%282%2F3%29%2Ax%29%29=7 Replace y with 14%2F3%2B%282%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B2%2A%2814%2F3%29%2B2%282%2F3%29x=7 Distribute 2 to 14%2F3%2B%282%2F3%29%2Ax

1%2Ax%2B28%2F3%2B%284%2F3%29%2Ax=7 Multiply



1%2Ax%2B28%2F3%2B%284%2F3%29%2Ax=7 Reduce any fractions

1%2Ax%2B%284%2F3%29%2Ax=7-28%2F3 Subtract 28%2F3 from both sides


1%2Ax%2B%284%2F3%29%2Ax=21%2F3-28%2F3 Make 7 into a fraction with a denominator of 3


1%2Ax%2B%284%2F3%29%2Ax=-7%2F3 Combine the terms on the right side



%283%2F3%29%2Ax%2B%284%2F3%29x=-7%2F3 Make 1 into a fraction with a denominator of 3

%287%2F3%29%2Ax=-7%2F3 Now combine the terms on the left side.


cross%28%283%2F7%29%287%2F3%29%29x=%28-7%2F3%29%283%2F7%29 Multiply both sides by 3%2F7. This will cancel out 7%2F3 and isolate x

So when we multiply -7%2F3 and 3%2F7 (and simplify) we get



x=-1 <---------------------------------One answer

Now that we know that x=-1, lets substitute that in for x to solve for y

1%28-1%29%2B2%2Ay=7 Plug in x=-1 into the 2nd equation

-1%2B2%2Ay=7 Multiply

2%2Ay=7%2B1Add 1 to both sides

2%2Ay=8 Combine the terms on the right side

cross%28%281%2F2%29%282%29%29%2Ay=%288%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.

y=8%2F2 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=-1 and y=4

which can also look like

(-1,4)

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

-2%2Ax%2B3%2Ay=14
1%2Ax%2B2%2Ay=7

we get


graph of -2%2Ax%2B3%2Ay=14 (red) and 1%2Ax%2B2%2Ay=7 (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 (-1,4). This verifies our answer.


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Check:

Plug in (-1,4) into the system of equations


Let x=-1 and y=4. Now plug those values into the equation -2%2Ax%2B3%2Ay=14

-2%2A%28-1%29%2B3%2A%284%29=14 Plug in x=-1 and y=4


2%2B12=14 Multiply


14=14 Add


14=14 Reduce. Since this equation is true the solution works.


So the solution (-1,4) satisfies -2%2Ax%2B3%2Ay=14



Let x=-1 and y=4. Now plug those values into the equation 1%2Ax%2B2%2Ay=7

1%2A%28-1%29%2B2%2A%284%29=7 Plug in x=-1 and y=4


-1%2B8=7 Multiply


7=7 Add


7=7 Reduce. Since this equation is true the solution works.


So the solution (-1,4) satisfies 1%2Ax%2B2%2Ay=7


Since the solution (-1,4) satisfies the system of equations


-2%2Ax%2B3%2Ay=14
1%2Ax%2B2%2Ay=7


this verifies our answer.