SOLUTION: 4x+3y=-22 x-3y=2

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Question 119960: 4x+3y=-22
x-3y=2
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

4%2Ax%2B3%2Ay=-22
1%2Ax-3%2Ay=2

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=-22-4%2AxSubtract 4%2Ax from both sides

y=%28-22-4%2Ax%29%2F3 Divide both sides by 3.


Which breaks down and reduces to



y=-22%2F3-%284%2F3%29%2Ax Now we've fully isolated y

Since y equals -22%2F3-%284%2F3%29%2Ax we can substitute the expression -22%2F3-%284%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-3%2Ahighlight%28%28-22%2F3-%284%2F3%29%2Ax%29%29=2 Replace y with -22%2F3-%284%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-3%2A%28-22%2F3%29-3%28-4%2F3%29x=2 Distribute -3 to -22%2F3-%284%2F3%29%2Ax

1%2Ax%2B66%2F3%2B%2812%2F3%29%2Ax=2 Multiply



1%2Ax%2B22%2B4%2Ax=2 Reduce any fractions

1%2Ax%2B4%2Ax=2-22 Subtract 22 from both sides


1%2Ax%2B4%2Ax=-20 Combine the terms on the right side



5%2Ax=-20 Now combine the terms on the left side.


cross%28%281%2F5%29%285%2F1%29%29x=%28-20%2F1%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5%2F1 and isolate x

So when we multiply -20%2F1 and 1%2F5 (and simplify) we get



x=-4 <---------------------------------One answer

Now that we know that x=-4, lets substitute that in for x to solve for y

1%28-4%29-3%2Ay=2 Plug in x=-4 into the 2nd equation

-4-3%2Ay=2 Multiply

-3%2Ay=2%2B4Add 4 to both sides

-3%2Ay=6 Combine the terms on the right side

cross%28%281%2F-3%29%28-3%29%29%2Ay=%286%2F1%29%281%2F-3%29 Multiply both sides by 1%2F-3. This will cancel out -3 on the left side.

y=6%2F-3 Multiply the terms on the right side


y=-2 Reduce


So this is the other answer


y=-2<---------------------------------Other answer


So our solution is

x=-4 and y=-2

which can also look like

(-4,-2)

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

4%2Ax%2B3%2Ay=-22
1%2Ax-3%2Ay=2

we get


graph of 4%2Ax%2B3%2Ay=-22 (red) and 1%2Ax-3%2Ay=2 (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 (-4,-2). This verifies our answer.


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Check:

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


Let x=-4 and y=-2. Now plug those values into the equation 4%2Ax%2B3%2Ay=-22

4%2A%28-4%29%2B3%2A%28-2%29=-22 Plug in x=-4 and y=-2


-16-6=-22 Multiply


-22=-22 Add


-22=-22 Reduce. Since this equation is true the solution works.


So the solution (-4,-2) satisfies 4%2Ax%2B3%2Ay=-22



Let x=-4 and y=-2. Now plug those values into the equation 1%2Ax-3%2Ay=2

1%2A%28-4%29-3%2A%28-2%29=2 Plug in x=-4 and y=-2


-4%2B6=2 Multiply


2=2 Add


2=2 Reduce. Since this equation is true the solution works.


So the solution (-4,-2) satisfies 1%2Ax-3%2Ay=2


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


4%2Ax%2B3%2Ay=-22
1%2Ax-3%2Ay=2


this verifies our answer.