SOLUTION: Solve each system: 3p+2q=-2 9p-q=-6

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Question 115037This question is from textbook
: Solve each system:
3p+2q=-2
9p-q=-6 This question is from textbook

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

3%2Ax%2B2%2Ay=-2
9%2Ax-1%2Ay=-6

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

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

y=%28-2-3%2Ax%29%2F2 Divide both sides by 2.


Which breaks down and reduces to



y=-1-%283%2F2%29%2Ax Now we've fully isolated y

Since y equals -1-%283%2F2%29%2Ax we can substitute the expression -1-%283%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


9%2Ax%2B-1%2Ahighlight%28%28-1-%283%2F2%29%2Ax%29%29=-6 Replace y with -1-%283%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

9%2Ax-1%2A%28-1%29-1%28-3%2F2%29x=-6 Distribute -1 to -1-%283%2F2%29%2Ax

9%2Ax%2B1%2B%283%2F2%29%2Ax=-6 Multiply



9%2Ax%2B1%2B%283%2F2%29%2Ax=-6 Reduce any fractions

9%2Ax%2B%283%2F2%29%2Ax=-6-1 Subtract 1 from both sides


9%2Ax%2B%283%2F2%29%2Ax=-7 Combine the terms on the right side



%2818%2F2%29%2Ax%2B%283%2F2%29x=-7 Make 9 into a fraction with a denominator of 2

%2821%2F2%29%2Ax=-7 Now combine the terms on the left side.


cross%28%282%2F21%29%2821%2F2%29%29x=%28-7%2F1%29%282%2F21%29 Multiply both sides by 2%2F21. This will cancel out 21%2F2 and isolate x

So when we multiply -7%2F1 and 2%2F21 (and simplify) we get



x=-2%2F3 <---------------------------------One answer

Now that we know that x=-2%2F3, lets substitute that in for x to solve for y

9%28-2%2F3%29-1%2Ay=-6 Plug in x=-2%2F3 into the 2nd equation

-6-1%2Ay=-6 Multiply

-1%2Ay=-6%2B6Add 6 to both sides

-1%2Ay=0 Combine the terms on the right side

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

y=0%2F-1 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=-2%2F3 and y=0

which can also look like

(-2%2F3,0)

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

3%2Ax%2B2%2Ay=-2
9%2Ax-1%2Ay=-6

we get


graph of 3%2Ax%2B2%2Ay=-2 (red) and 9%2Ax-1%2Ay=-6 (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%2F3,0). This verifies our answer.


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Check:

Plug in (-2%2F3,0) into the system of equations


Let x=-2%2F3 and y=0. Now plug those values into the equation 3%2Ax%2B2%2Ay=-2

3%2A%28-2%2F3%29%2B2%2A%280%29=-2 Plug in x=-2%2F3 and y=0


-6%2F3%2B0=-2 Multiply


-6%2F3=-2 Add


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


So the solution (-2%2F3,0) satisfies 3%2Ax%2B2%2Ay=-2



Let x=-2%2F3 and y=0. Now plug those values into the equation 9%2Ax-1%2Ay=-6

9%2A%28-2%2F3%29-1%2A%280%29=-6 Plug in x=-2%2F3 and y=0


-18%2F3%2B0=-6 Multiply


-18%2F3=-6 Add


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


So the solution (-2%2F3,0) satisfies 9%2Ax-1%2Ay=-6


Since the solution (-2%2F3,0) satisfies the system of equations


3%2Ax%2B2%2Ay=-2
9%2Ax-1%2Ay=-6


this verifies our answer.