SOLUTION: Help me please i dont understand 2x+2/3y=-8 1/2x-1/3y=1

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Question 96281: Help me please i dont understand
2x+2/3y=-8
1/2x-1/3y=1
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


%282%29%2Ax%2B%282%2F3%29%2Ay=-8 Start with the first equation


3%28%282%29%2Ax%2B%282%2F3%29%2Ay%29=%283%29%2A%28-8%29 Multiply both sides by the LCD 3



6%2Ax%2B2%2Ay=-24Distribute and simplify


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%281%2F2%29%2Ax%2B%28-1%2F3%29%2Ay=1 Start with the second equation


6%28%281%2F2%29%2Ax%2B%28-1%2F3%29%2Ay%29=%286%29%2A%281%29 Multiply both sides by the LCD 6



3%2Ax%2B-2%2Ay=6 Distribute and simplify


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Lets start with the given system of linear equations

6%2Ax%2B2%2Ay=-24
3%2Ax-2%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=-24-6%2AxSubtract 6%2Ax from both sides

y=%28-24-6%2Ax%29%2F2 Divide both sides by 2.


Which breaks down and reduces to



y=-12-3%2Ax Now we've fully isolated y

Since y equals -12-3%2Ax we can substitute the expression -12-3%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B-2%2Ahighlight%28%28-12-3%2Ax%29%29=6 Replace y with -12-3%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax-2%2A%28-12%29-2%28-3%29x=6 Distribute -2 to -12-3%2Ax

3%2Ax%2B24%2B6%2Ax=6 Multiply



3%2Ax%2B24%2B6%2Ax=6 Reduce any fractions

3%2Ax%2B6%2Ax=6-24 Subtract 24 from both sides


3%2Ax%2B6%2Ax=-18 Combine the terms on the right side



9%2Ax=-18 Now combine the terms on the left side.


cross%28%281%2F9%29%289%2F1%29%29x=%28-18%2F1%29%281%2F9%29 Multiply both sides by 1%2F9. This will cancel out 9%2F1 and isolate x

So when we multiply -18%2F1 and 1%2F9 (and simplify) we get



x=-2 <---------------------------------One answer

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

3%28-2%29-2%2Ay=6 Plug in x=-2 into the 2nd equation

-6-2%2Ay=6 Multiply

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

-2%2Ay=12 Combine the terms on the right side

cross%28%281%2F-2%29%28-2%29%29%2Ay=%2812%2F1%29%281%2F-2%29 Multiply both sides by 1%2F-2. This will cancel out -2 on the left side.

y=12%2F-2 Multiply the terms on the right side


y=-6 Reduce


So this is the other answer


y=-6<---------------------------------Other answer


So our solution is

x=-2 and y=-6

which can also look like

(-2,-6)

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

6%2Ax%2B2%2Ay=-24
3%2Ax-2%2Ay=6

we get


graph of 6%2Ax%2B2%2Ay=-24 (red) and 3%2Ax-2%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,-6). This verifies our answer.


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Check:

Plug in (-2,-6) into the system of equations


Let x=-2 and y=-6. Now plug those values into the equation 6%2Ax%2B2%2Ay=-24

6%2A%28-2%29%2B2%2A%28-6%29=-24 Plug in x=-2 and y=-6


-12-12=-24 Multiply


-24=-24 Add


-24=-24 Reduce. Since this equation is true the solution works.


So the solution (-2,-6) satisfies 6%2Ax%2B2%2Ay=-24



Let x=-2 and y=-6. Now plug those values into the equation 3%2Ax-2%2Ay=6

3%2A%28-2%29-2%2A%28-6%29=6 Plug in x=-2 and y=-6


-6%2B12=6 Multiply


6=6 Add


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


So the solution (-2,-6) satisfies 3%2Ax-2%2Ay=6


Since the solution (-2,-6) satisfies the system of equations


6%2Ax%2B2%2Ay=-24
3%2Ax-2%2Ay=6


this verifies our answer.