SOLUTION: 35. solve each systems of equations algebraically x+8y=0 -x+y=-9

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Question 120662: 35. solve each systems of equations algebraically

x+8y=0
-x+y=-9
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

#35

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


Lets start with the given system of linear equations

1%2Ax%2B8%2Ay=0
-1%2Ax%2B1%2Ay=-9

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

8%2Ay=0-1%2AxSubtract 1%2Ax from both sides

y=%280-1%2Ax%29%2F8 Divide both sides by 8.


Which breaks down and reduces to



y=0-%281%2F8%29%2Ax Now we've fully isolated y

Since y equals 0-%281%2F8%29%2Ax we can substitute the expression 0-%281%2F8%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-1%2Ax%2B1%2Ahighlight%28%280-%281%2F8%29%2Ax%29%29=-9 Replace y with 0-%281%2F8%29%2Ax. Since this eliminates y, we can now solve for x.

-1%2Ax%2B1%2A%280%29%2B1%28-1%2F8%29x=-9 Distribute 1 to 0-%281%2F8%29%2Ax

-1%2Ax%2B0-%281%2F8%29%2Ax=-9 Multiply



-1%2Ax%2B0-%281%2F8%29%2Ax=-9 Reduce any fractions

-1%2Ax-%281%2F8%29%2Ax=-9%2B0Add 0 to both sides


-1%2Ax-%281%2F8%29%2Ax=-9 Combine the terms on the right side



%28-8%2F8%29%2Ax-%281%2F8%29x=-9 Make -1 into a fraction with a denominator of 8

%28-9%2F8%29%2Ax=-9 Now combine the terms on the left side.


cross%28%288%2F-9%29%28-9%2F8%29%29x=%28-9%2F1%29%288%2F-9%29 Multiply both sides by 8%2F-9. This will cancel out -9%2F8 and isolate x

So when we multiply -9%2F1 and 8%2F-9 (and simplify) we get



x=8 <---------------------------------One answer

Now that we know that x=8, lets substitute that in for x to solve for y

-1%288%29%2B1%2Ay=-9 Plug in x=8 into the 2nd equation

-8%2B1%2Ay=-9 Multiply

1%2Ay=-9%2B8Add 8 to both sides

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

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

y=-1%2F1 Multiply the terms on the right side


y=-1 Reduce


So this is the other answer


y=-1<---------------------------------Other answer


So our solution is

x=8 and y=-1

which can also look like

(8,-1)

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

1%2Ax%2B8%2Ay=0
-1%2Ax%2B1%2Ay=-9

we get


graph of 1%2Ax%2B8%2Ay=0 (red) and -1%2Ax%2B1%2Ay=-9 (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 (8,-1). This verifies our answer.


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Check:

Plug in (8,-1) into the system of equations


Let x=8 and y=-1. Now plug those values into the equation 1%2Ax%2B8%2Ay=0

1%2A%288%29%2B8%2A%28-1%29=0 Plug in x=8 and y=-1


8-8=0 Multiply


0=0 Add


0=0 Reduce. Since this equation is true the solution works.


So the solution (8,-1) satisfies 1%2Ax%2B8%2Ay=0



Let x=8 and y=-1. Now plug those values into the equation -1%2Ax%2B1%2Ay=-9

-1%2A%288%29%2B1%2A%28-1%29=-9 Plug in x=8 and y=-1


-8-1=-9 Multiply


-9=-9 Add


-9=-9 Reduce. Since this equation is true the solution works.


So the solution (8,-1) satisfies -1%2Ax%2B1%2Ay=-9


Since the solution (8,-1) satisfies the system of equations


1%2Ax%2B8%2Ay=0
-1%2Ax%2B1%2Ay=-9


this verifies our answer.