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.

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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) About Me  (Show Source):
You can put this solution on YOUR website!
Start with the given system
5x-2y=-5
y-5x=3

5x-2y=-5
-5x%2By=3 Rearrange the terms
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
-5%2Ax%2B1%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.


-5%2Ax%2B1%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.

-5%2Ax%2B1%2A%285%2F2%29%2B1%285%2F2%29x=3 Distribute 1 to 5%2F2%2B%285%2F2%29%2Ax

-5%2Ax%2B5%2F2%2B%285%2F2%29%2Ax=3 Multiply



-5%2Ax%2B5%2F2%2B%285%2F2%29%2Ax=3 Reduce any fractions

-5%2Ax%2B%285%2F2%29%2Ax=3-5%2F2 Subtract 5%2F2 from both sides


-5%2Ax%2B%285%2F2%29%2Ax=6%2F2-5%2F2 Make 3 into a fraction with a denominator of 2


-5%2Ax%2B%285%2F2%29%2Ax=1%2F2 Combine the terms on the right side



%28-10%2F2%29%2Ax%2B%285%2F2%29x=1%2F2 Make -5 into a fraction with a denominator of 2

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


cross%28%282%2F-5%29%28-5%2F2%29%29x=%281%2F2%29%282%2F-5%29 Multiply both sides by 2%2F-5. This will cancel out -5%2F2 and isolate x

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



x=-1%2F5 <---------------------------------One answer

Now that we know that x=-1%2F5, lets substitute that in for x to solve for y

-5%28-1%2F5%29%2B1%2Ay=3 Plug in x=-1%2F5 into the 2nd equation

1%2B1%2Ay=3 Multiply

1%2Ay=3-1Subtract 1 from both sides

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

cross%28%281%2F1%29%281%29%29%2Ay=%282%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=2%2F1 Multiply the terms on the right side


y=2 Reduce


So this is the other answer


y=2<---------------------------------Other answer


So our solution is

x=-1%2F5 and y=2

which can also look like

(-1%2F5,2)

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

5%2Ax-2%2Ay=-5
-5%2Ax%2B1%2Ay=3

we get


graph of 5%2Ax-2%2Ay=-5 (red) and -5%2Ax%2B1%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 (-1%2F5,2). This verifies our answer.


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

Plug in (-1%2F5,2) into the system of equations


Let x=-1%2F5 and y=2. Now plug those values into the equation 5%2Ax-2%2Ay=-5

5%2A%28-1%2F5%29-2%2A%282%29=-5 Plug in x=-1%2F5 and y=2


-5%2F5-4=-5 Multiply


-25%2F5=-5 Add


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


So the solution (-1%2F5,2) satisfies 5%2Ax-2%2Ay=-5



Let x=-1%2F5 and y=2. Now plug those values into the equation -5%2Ax%2B1%2Ay=3

-5%2A%28-1%2F5%29%2B1%2A%282%29=3 Plug in x=-1%2F5 and y=2


5%2F5%2B2=3 Multiply


15%2F5=3 Add


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


So the solution (-1%2F5,2) satisfies -5%2Ax%2B1%2Ay=3


Since the solution (-1%2F5,2) satisfies the system of equations


5%2Ax-2%2Ay=-5
-5%2Ax%2B1%2Ay=3


this verifies our answer.




----------------------------------------------------------------------------
Start with the given system
8x-4y=16
y=2x-4+

8x-4%282x-4%29=16 Plug in y=2x-4
8x-8x%2B16=16 Distribute
16=16 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

4%2Ax-12%2Ay=5
-1%2Ax%2B3%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

-12%2Ay=5-4%2AxSubtract 4%2Ax from both sides

y=%285-4%2Ax%29%2F-12 Divide both sides by -12.


Which breaks down and reduces to



y=-5%2F12%2B%281%2F3%29%2Ax Now we've fully isolated y

Since y equals -5%2F12%2B%281%2F3%29%2Ax we can substitute the expression -5%2F12%2B%281%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-1%2Ax%2B3%2Ahighlight%28%28-5%2F12%2B%281%2F3%29%2Ax%29%29=-1 Replace y with -5%2F12%2B%281%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

-1%2Ax%2B3%2A%28-5%2F12%29%2B3%281%2F3%29x=-1 Distribute 3 to -5%2F12%2B%281%2F3%29%2Ax

-1%2Ax-15%2F12%2B%283%2F3%29%2Ax=-1 Multiply



-1%2Ax-5%2F4%2B1%2Ax=-1 Reduce any fractions

-1%2Ax%2B1%2Ax=-1%2B5%2F4Add 5%2F4 to both sides


-1%2Ax%2B1%2Ax=-4%2F4%2B5%2F4 Make -1 into a fraction with a denominator of 4


-1%2Ax%2B1%2Ax=1%2F4 Combine the terms on the right side



0%2Ax=1%2F4 Now combine the terms on the left side.
0%2F1=1%2F4 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 4%2Ax-12%2Ay=5 (red) and -1%2Ax%2B3%2Ay=-1 (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