SOLUTION: Solve the following systems by substitution. 5x - 2y=-5 y- 5x =3

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Question 91969: Solve the following systems by substitution.
5x - 2y=-5
y- 5x =3
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

5%2Ax-2%2Ay=-5
1%2Ax-5%2Ay=3

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=-5-5%2AxSubtract 5%2Ax from both sides

y=%28-5-5%2Ax%29%2F-2 Divide both sides by -2.


Which breaks down and reduces to



y=5%2F2%2B%285%2F2%29%2Ax Now we've fully isolated y

Since y equals 5%2F2%2B%285%2F2%29%2Ax we can substitute the expression 5%2F2%2B%285%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-5%2Ahighlight%28%285%2F2%2B%285%2F2%29%2Ax%29%29=3 Replace y with 5%2F2%2B%285%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-5%2A%285%2F2%29-5%285%2F2%29x=3 Distribute -5 to 5%2F2%2B%285%2F2%29%2Ax

1%2Ax-25%2F2-%2825%2F2%29%2Ax=3 Multiply



1%2Ax-25%2F2-%2825%2F2%29%2Ax=3 Reduce any fractions

1%2Ax-%2825%2F2%29%2Ax=3%2B25%2F2Add 25%2F2 to both sides


1%2Ax-%2825%2F2%29%2Ax=6%2F2%2B25%2F2 Make 3 into a fraction with a denominator of 2


1%2Ax-%2825%2F2%29%2Ax=31%2F2 Combine the terms on the right side



%282%2F2%29%2Ax-%2825%2F2%29x=31%2F2 Make 1 into a fraction with a denominator of 2

%28-23%2F2%29%2Ax=31%2F2 Now combine the terms on the left side.


cross%28%282%2F-23%29%28-23%2F2%29%29x=%2831%2F2%29%282%2F-23%29 Multiply both sides by 2%2F-23. This will cancel out -23%2F2 and isolate x

So when we multiply 31%2F2 and 2%2F-23 (and simplify) we get



x=-31%2F23 <---------------------------------One answer

Now that we know that x=-31%2F23, lets substitute that in for x to solve for y

1%28-31%2F23%29-5%2Ay=3 Plug in x=-31%2F23 into the 2nd equation

-31%2F23-5%2Ay=3 Multiply

-5%2Ay=3%2B31%2F23Add 31%2F23 to both sides

-5%2Ay=69%2F23%2B31%2F23 Make 3 into a fraction with a denominator of 23



-5%2Ay=100%2F23 Combine the terms on the right side

cross%28%281%2F-5%29%28-5%29%29%2Ay=%28100%2F23%29%281%2F-5%29 Multiply both sides by 1%2F-5. This will cancel out -5 on the left side.

y=100%2F-115 Multiply the terms on the right side


y=-20%2F23 Reduce


So this is the other answer


y=-20%2F23<---------------------------------Other answer


So our solution is

x=-31%2F23 and y=-20%2F23

which can also look like

(-31%2F23,-20%2F23)

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

5%2Ax-2%2Ay=-5
1%2Ax-5%2Ay=3

we get


graph of 5%2Ax-2%2Ay=-5 (red) and 1%2Ax-5%2Ay=3 (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 (-31%2F23,-20%2F23). This verifies our answer.


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Check:

Plug in (-31%2F23,-20%2F23) into the system of equations


Let x=-31%2F23 and y=-20%2F23. Now plug those values into the equation 5%2Ax-2%2Ay=-5

5%2A%28-31%2F23%29-2%2A%28-20%2F23%29=-5 Plug in x=-31%2F23 and y=-20%2F23


-155%2F23%2B40%2F23=-5 Multiply


-115%2F23=-5 Add


-5=-5 Reduce. Since this equation is true the solution works.


So the solution (-31%2F23,-20%2F23) satisfies 5%2Ax-2%2Ay=-5



Let x=-31%2F23 and y=-20%2F23. Now plug those values into the equation 1%2Ax-5%2Ay=3

1%2A%28-31%2F23%29-5%2A%28-20%2F23%29=3 Plug in x=-31%2F23 and y=-20%2F23


-31%2F23%2B100%2F23=3 Multiply


69%2F23=3 Add


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


So the solution (-31%2F23,-20%2F23) satisfies 1%2Ax-5%2Ay=3


Since the solution (-31%2F23,-20%2F23) satisfies the system of equations


5%2Ax-2%2Ay=-5
1%2Ax-5%2Ay=3


this verifies our answer.