SOLUTION: please help me to solve the systems of equations 3/2X-1/3Y=1/2 2X-1/2Y=-1/2

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Question 92524: please help me to solve the systems of equations

3/2X-1/3Y=1/2
2X-1/2Y=-1/2
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


%283%2F2%29%2Ax%2B%28-1%2F3%29%2Ay=1%2F2 Start with the first equation


6%28%283%2F2%29%2Ax%2B%28-1%2F3%29%2Ay%29=%286%29%2A%281%2F2%29 Multiply both sides by the LCD 6



9%2Ax%2B-2%2Ay=3Distribute and simplify


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%282%29%2Ax%2B%28-1%2F2%29%2Ay=-1%2F2 Start with the second equation


2%28%282%29%2Ax%2B%28-1%2F2%29%2Ay%29=%282%29%2A%28-1%2F2%29 Multiply both sides by the LCD 2



4%2Ax%2B-1%2Ay=-1 Distribute and simplify


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Lets start with the given system of linear equations

9%2Ax-2%2Ay=3
4%2Ax-1%2Ay=-1

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=3-9%2AxSubtract 9%2Ax from both sides

y=%283-9%2Ax%29%2F-2 Divide both sides by -2.


Which breaks down and reduces to



y=-3%2F2%2B%289%2F2%29%2Ax Now we've fully isolated y

Since y equals -3%2F2%2B%289%2F2%29%2Ax we can substitute the expression -3%2F2%2B%289%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


4%2Ax%2B-1%2Ahighlight%28%28-3%2F2%2B%289%2F2%29%2Ax%29%29=-1 Replace y with -3%2F2%2B%289%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

4%2Ax-1%2A%28-3%2F2%29-1%289%2F2%29x=-1 Distribute -1 to -3%2F2%2B%289%2F2%29%2Ax

4%2Ax%2B3%2F2-%289%2F2%29%2Ax=-1 Multiply



4%2Ax%2B3%2F2-%289%2F2%29%2Ax=-1 Reduce any fractions

4%2Ax-%289%2F2%29%2Ax=-1-3%2F2 Subtract 3%2F2 from both sides


4%2Ax-%289%2F2%29%2Ax=-2%2F2-3%2F2 Make -1 into a fraction with a denominator of 2


4%2Ax-%289%2F2%29%2Ax=-5%2F2 Combine the terms on the right side



%288%2F2%29%2Ax-%289%2F2%29x=-5%2F2 Make 4 into a fraction with a denominator of 2

%28-1%2F2%29%2Ax=-5%2F2 Now combine the terms on the left side.


cross%28%282%2F-1%29%28-1%2F2%29%29x=%28-5%2F2%29%282%2F-1%29 Multiply both sides by 2%2F-1. This will cancel out -1%2F2 and isolate x

So when we multiply -5%2F2 and 2%2F-1 (and simplify) we get



x=5 <---------------------------------One answer

Now that we know that x=5, lets substitute that in for x to solve for y

4%285%29-1%2Ay=-1 Plug in x=5 into the 2nd equation

20-1%2Ay=-1 Multiply

-1%2Ay=-1-20Subtract 20 from both sides

-1%2Ay=-21 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ay=%28-21%2F1%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.

y=-21%2F-1 Multiply the terms on the right side


y=21 Reduce


So this is the other answer


y=21<---------------------------------Other answer


So our solution is

x=5 and y=21

which can also look like

(5,21)

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

9%2Ax-2%2Ay=3
4%2Ax-1%2Ay=-1

we get


graph of 9%2Ax-2%2Ay=3 (red) and 4%2Ax-1%2Ay=-1 (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 (5,21). This verifies our answer.


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Check:

Plug in (5,21) into the system of equations


Let x=5 and y=21. Now plug those values into the equation 9%2Ax-2%2Ay=3

9%2A%285%29-2%2A%2821%29=3 Plug in x=5 and y=21


45-42=3 Multiply


3=3 Add


3=3 Reduce. Since this equation is true the solution works.


So the solution (5,21) satisfies 9%2Ax-2%2Ay=3



Let x=5 and y=21. Now plug those values into the equation 4%2Ax-1%2Ay=-1

4%2A%285%29-1%2A%2821%29=-1 Plug in x=5 and y=21


20-21=-1 Multiply


-1=-1 Add


-1=-1 Reduce. Since this equation is true the solution works.


So the solution (5,21) satisfies 4%2Ax-1%2Ay=-1


Since the solution (5,21) satisfies the system of equations


9%2Ax-2%2Ay=3
4%2Ax-1%2Ay=-1


this verifies our answer.