SOLUTION: Solve the following system of equations 1/3x -1/4y =-6 2x - 1/5y =2/5

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Question 1008442: Solve the following system of equations
1/3x -1/4y =-6
2x - 1/5y =2/5
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
%281%2F3%29x+-%281%2F4%29y+=-6
2x+-+%281%2F5%29y+=2%2F5
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Solved by pluggable solver: Solving a linear system of equations by subsitution


%281%2F3%29%2Ax%2B%28-1%2F4%29%2Ay=-6 Start with the first equation


12%28%281%2F3%29%2Ax%2B%28-1%2F4%29%2Ay%29=%2812%29%2A%28-6%29 Multiply both sides by the LCD 12



4%2Ax%2B-3%2Ay=-72Distribute and simplify


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%282%29%2Ax%2B%28-1%2F5%29%2Ay=2%2F5 Start with the second equation


5%28%282%29%2Ax%2B%28-1%2F5%29%2Ay%29=%285%29%2A%282%2F5%29 Multiply both sides by the LCD 5



10%2Ax%2B-1%2Ay=2 Distribute and simplify


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Lets start with the given system of linear equations

4%2Ax-3%2Ay=-72
10%2Ax-1%2Ay=2

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

-3%2Ay=-72-4%2AxSubtract 4%2Ax from both sides

y=%28-72-4%2Ax%29%2F-3 Divide both sides by -3.


Which breaks down and reduces to



y=24%2B%284%2F3%29%2Ax Now we've fully isolated y

Since y equals 24%2B%284%2F3%29%2Ax we can substitute the expression 24%2B%284%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


10%2Ax%2B-1%2Ahighlight%28%2824%2B%284%2F3%29%2Ax%29%29=2 Replace y with 24%2B%284%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

10%2Ax-1%2A%2824%29-1%284%2F3%29x=2 Distribute -1 to 24%2B%284%2F3%29%2Ax

10%2Ax-24-%284%2F3%29%2Ax=2 Multiply



10%2Ax-24-%284%2F3%29%2Ax=2 Reduce any fractions

10%2Ax-%284%2F3%29%2Ax=2%2B24Add 24 to both sides


10%2Ax-%284%2F3%29%2Ax=26 Combine the terms on the right side



%2830%2F3%29%2Ax-%284%2F3%29x=26 Make 10 into a fraction with a denominator of 3

%2826%2F3%29%2Ax=26 Now combine the terms on the left side.


cross%28%283%2F26%29%2826%2F3%29%29x=%2826%2F1%29%283%2F26%29 Multiply both sides by 3%2F26. This will cancel out 26%2F3 and isolate x

So when we multiply 26%2F1 and 3%2F26 (and simplify) we get



x=3 <---------------------------------One answer

Now that we know that x=3, lets substitute that in for x to solve for y

10%283%29-1%2Ay=2 Plug in x=3 into the 2nd equation

30-1%2Ay=2 Multiply

-1%2Ay=2-30Subtract 30 from both sides

-1%2Ay=-28 Combine the terms on the right side

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

y=-28%2F-1 Multiply the terms on the right side


y=28 Reduce


So this is the other answer


y=28<---------------------------------Other answer


So our solution is

x=3 and y=28

which can also look like

(3,28)

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

4%2Ax-3%2Ay=-72
10%2Ax-1%2Ay=2

we get


graph of 4%2Ax-3%2Ay=-72 (red) and 10%2Ax-1%2Ay=2 (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 (3,28). This verifies our answer.


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Check:

Plug in (3,28) into the system of equations


Let x=3 and y=28. Now plug those values into the equation 4%2Ax-3%2Ay=-72

4%2A%283%29-3%2A%2828%29=-72 Plug in x=3 and y=28


12-84=-72 Multiply


-72=-72 Add


-72=-72 Reduce. Since this equation is true the solution works.


So the solution (3,28) satisfies 4%2Ax-3%2Ay=-72



Let x=3 and y=28. Now plug those values into the equation 10%2Ax-1%2Ay=2

10%2A%283%29-1%2A%2828%29=2 Plug in x=3 and y=28


30-28=2 Multiply


2=2 Add


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


So the solution (3,28) satisfies 10%2Ax-1%2Ay=2


Since the solution (3,28) satisfies the system of equations


4%2Ax-3%2Ay=-72
10%2Ax-1%2Ay=2


this verifies our answer.

Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!

Solve the following system of equations
1/3x -1/4y =-6
2x - 1/5y =2/5
%281%2F3%29x+-+%281%2F4%29y+=+-+6 

x%2F3+-+y%2F4+=+-+6

4x - 3y = - 72 -------- Multiplying by LCD, 12 ------ eq (i)



2x+-+%281%2F5%29y+=+2%2F5 

2x+-+y%2F5+=+2%2F5

10x - y = 2 --------- Multiplying by LCD, 5 ---------- eq (ii)

- 30x + 3y = - 6 ---- Multiplying eq (ii) by - 3 ----- eq (iii)

- 26x = - 78 -------- Adding eqs (iii) & (i)

x = %28-+78%29%2F%28-+26%29, or highlight_green%283%29



4(3) - 3y = - 72 ------ Substituting 3 for x in eq (i)

12 - 3y = - 72

- 3y = - 72 - 12

- 3y = - 84

y = %28-+84%29%2F%28-+3%29, or highlight_green%2828%29