SOLUTION: Solve each system of equations. Check by substitution 3a + 4b = -12 4a - 5b = -16

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Question 120289: Solve each system of equations. Check by substitution
3a + 4b = -12
4a - 5b = -16
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

1.
3a+%2B+4b+=+-12
4a+-+5b+=+-16
solution:
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

3%2Ax%2B4%2Ay=-12
4%2Ax-5%2Ay=-16

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=-12-3%2AxSubtract 3%2Ax from both sides

y=%28-12-3%2Ax%29%2F4 Divide both sides by 4.


Which breaks down and reduces to



y=-3-%283%2F4%29%2Ax Now we've fully isolated y

Since y equals -3-%283%2F4%29%2Ax we can substitute the expression -3-%283%2F4%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


4%2Ax%2B-5%2Ahighlight%28%28-3-%283%2F4%29%2Ax%29%29=-16 Replace y with -3-%283%2F4%29%2Ax. Since this eliminates y, we can now solve for x.

4%2Ax-5%2A%28-3%29-5%28-3%2F4%29x=-16 Distribute -5 to -3-%283%2F4%29%2Ax

4%2Ax%2B15%2B%2815%2F4%29%2Ax=-16 Multiply



4%2Ax%2B15%2B%2815%2F4%29%2Ax=-16 Reduce any fractions

4%2Ax%2B%2815%2F4%29%2Ax=-16-15 Subtract 15 from both sides


4%2Ax%2B%2815%2F4%29%2Ax=-31 Combine the terms on the right side



%2816%2F4%29%2Ax%2B%2815%2F4%29x=-31 Make 4 into a fraction with a denominator of 4

%2831%2F4%29%2Ax=-31 Now combine the terms on the left side.


cross%28%284%2F31%29%2831%2F4%29%29x=%28-31%2F1%29%284%2F31%29 Multiply both sides by 4%2F31. This will cancel out 31%2F4 and isolate x

So when we multiply -31%2F1 and 4%2F31 (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

4%28-4%29-5%2Ay=-16 Plug in x=-4 into the 2nd equation

-16-5%2Ay=-16 Multiply

-5%2Ay=-16%2B16Add 16 to both sides

-5%2Ay=0 Combine the terms on the right side

cross%28%281%2F-5%29%28-5%29%29%2Ay=%280%2F1%29%281%2F-5%29 Multiply both sides by 1%2F-5. This will cancel out -5 on the left side.

y=0%2F-5 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)

3%2Ax%2B4%2Ay=-12
4%2Ax-5%2Ay=-16

we get


graph of 3%2Ax%2B4%2Ay=-12 (red) and 4%2Ax-5%2Ay=-16 (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 3%2Ax%2B4%2Ay=-12

3%2A%28-4%29%2B4%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%2B4%2Ay=-12



Let x=-4 and y=0. Now plug those values into the equation 4%2Ax-5%2Ay=-16

4%2A%28-4%29-5%2A%280%29=-16 Plug in x=-4 and y=0


-16%2B0=-16 Multiply


-16=-16 Add


-16=-16 Reduce. Since this equation is true the solution works.


So the solution (-4,0) satisfies 4%2Ax-5%2Ay=-16


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


3%2Ax%2B4%2Ay=-12
4%2Ax-5%2Ay=-16


this verifies our answer.