SOLUTION: Solve the system by substitution. x + 4y = -4 3x - 3y = -12 Thanks.

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Question 117686: Solve the system by substitution.
x + 4y = -4
3x - 3y = -12
Thanks.
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%2B4%2Ay=-4
3%2Ax-3%2Ay=-12

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

4%2Ay=-4-1%2AxSubtract 1%2Ax from both sides

y=%28-4-1%2Ax%29%2F4 Divide both sides by 4.


Which breaks down and reduces to



y=-1-%281%2F4%29%2Ax Now we've fully isolated y

Since y equals -1-%281%2F4%29%2Ax we can substitute the expression -1-%281%2F4%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B-3%2Ahighlight%28%28-1-%281%2F4%29%2Ax%29%29=-12 Replace y with -1-%281%2F4%29%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax-3%2A%28-1%29-3%28-1%2F4%29x=-12 Distribute -3 to -1-%281%2F4%29%2Ax

3%2Ax%2B3%2B%283%2F4%29%2Ax=-12 Multiply



3%2Ax%2B3%2B%283%2F4%29%2Ax=-12 Reduce any fractions

3%2Ax%2B%283%2F4%29%2Ax=-12-3 Subtract 3 from both sides


3%2Ax%2B%283%2F4%29%2Ax=-15 Combine the terms on the right side



%2812%2F4%29%2Ax%2B%283%2F4%29x=-15 Make 3 into a fraction with a denominator of 4

%2815%2F4%29%2Ax=-15 Now combine the terms on the left side.


cross%28%284%2F15%29%2815%2F4%29%29x=%28-15%2F1%29%284%2F15%29 Multiply both sides by 4%2F15. This will cancel out 15%2F4 and isolate x

So when we multiply -15%2F1 and 4%2F15 (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

3%28-4%29-3%2Ay=-12 Plug in x=-4 into the 2nd equation

-12-3%2Ay=-12 Multiply

-3%2Ay=-12%2B12Add 12 to both sides

-3%2Ay=0 Combine the terms on the right side

cross%28%281%2F-3%29%28-3%29%29%2Ay=%280%2F1%29%281%2F-3%29 Multiply both sides by 1%2F-3. This will cancel out -3 on the left side.

y=0%2F-3 Multiply the terms on the right side


y=0 Reduce


So this is the other answer


y=0<---------------------------------Other answer


So our solution is

x=-4 and y=0

which can also look like

(-4,0)

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

1%2Ax%2B4%2Ay=-4
3%2Ax-3%2Ay=-12

we get


graph of 1%2Ax%2B4%2Ay=-4 (red) and 3%2Ax-3%2Ay=-12 (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 (-4,0). This verifies our answer.


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Check:

Plug in (-4,0) into the system of equations


Let x=-4 and y=0. Now plug those values into the equation 1%2Ax%2B4%2Ay=-4

1%2A%28-4%29%2B4%2A%280%29=-4 Plug in x=-4 and y=0


-4%2B0=-4 Multiply


-4=-4 Add


-4=-4 Reduce. Since this equation is true the solution works.


So the solution (-4,0) satisfies 1%2Ax%2B4%2Ay=-4



Let x=-4 and y=0. Now plug those values into the equation 3%2Ax-3%2Ay=-12

3%2A%28-4%29-3%2A%280%29=-12 Plug in x=-4 and y=0


-12%2B0=-12 Multiply


-12=-12 Add


-12=-12 Reduce. Since this equation is true the solution works.


So the solution (-4,0) satisfies 3%2Ax-3%2Ay=-12


Since the solution (-4,0) satisfies the system of equations


1%2Ax%2B4%2Ay=-4
3%2Ax-3%2Ay=-12


this verifies our answer.