SOLUTION: I'm learning about these systems and I'm not understanding how to do this. Is it possible I can get some detail on how to solve tese? Solve these systyms with substitution.

Question 84073: I'm learning about these systems and I'm not understanding how to do this. Is it possible I can get some detail on how to solve tese?
Solve these systyms with substitution.
5x-2y=-5
y-5x=3

8x-4y=16
y=2x-4
4x-12y=5
-x+3y=-1
Thank you.

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Start with the given system

Rearrange the terms

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

Lets start with the given system of linear equations

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

Subtract from both sides

Divide both sides by -2.

Which breaks down and reduces to

Now we've fully isolated y

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

Replace y with . Since this eliminates y, we can now solve for x.

Distribute 1 to

Multiply

Reduce any fractions

Subtract from both sides

Make 3 into a fraction with a denominator of 2

Combine the terms on the right side

Make -5 into a fraction with a denominator of 2

Now combine the terms on the left side.

Multiply both sides by . This will cancel out and isolate x

So when we multiply and (and simplify) we get

<---------------------------------One answer

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

Plug in into the 2nd equation

Multiply

Subtract from both sides

Combine the terms on the right side

Multiply both sides by . This will cancel out 1 on the left side.

Multiply the terms on the right side

Reduce

So this is the other answer

<---------------------------------Other answer

So our solution is

and

which can also look like

(,)

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

we get

graph of (red) and (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 (,). This verifies our answer.

-----------------------------------------------------------------------------------------------
Check:

Plug in (,) into the system of equations

Let and . Now plug those values into the equation

Plug in and

Multiply

Add

Reduce. Since this equation is true the solution works.

So the solution (,) satisfies

Let and . Now plug those values into the equation

Plug in and

Multiply

Add

Reduce. Since this equation is true the solution works.

So the solution (,) satisfies

Since the solution (,) satisfies the system of equations

this verifies our answer.

----------------------------------------------------------------------------
Start with the given system

Plug in
Distribute
Combine like terms. Since this is true for any x or y, the system has an infinite number of solutions
----------------------------------------------------------------------------

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

Lets start with the given system of linear equations

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

Subtract from both sides

Divide both sides by -12.

Which breaks down and reduces to

Now we've fully isolated y

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

Replace y with . Since this eliminates y, we can now solve for x.

Distribute 3 to

Multiply

Reduce any fractions

Add to both sides

Make -1 into a fraction with a denominator of 4

Combine the terms on the right side

Now combine the terms on the left side.
Since this expression is not true, we have an inconsistency.

So there are no solutions. The simple reason is the 2 equations represent 2 parallel lines that will never intersect. Since no intersections occur, no solutions exist.

graph of (red) and (green) (hint: you may have to solve for y to graph these)

and we can see that the two equations are parallel and will never intersect. So this system is inconsistent

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