SOLUTION: I don't really understand how to do the substitution method. How would you answer {{{2x-3y=-25 3x+y=1}}}

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Question 120133: I don't really understand how to do the substitution method.
How would you answer 2x-3y=-25+%0D%0A++++++++++++++++++++++++3x%2By=1
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

2%2Ax-3%2Ay=-25
3%2Ax%2B1%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

-3%2Ay=-25-2%2AxSubtract 2%2Ax from both sides

y=%28-25-2%2Ax%29%2F-3 Divide both sides by -3.


Which breaks down and reduces to



y=25%2F3%2B%282%2F3%29%2Ax Now we've fully isolated y

Since y equals 25%2F3%2B%282%2F3%29%2Ax we can substitute the expression 25%2F3%2B%282%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B1%2Ahighlight%28%2825%2F3%2B%282%2F3%29%2Ax%29%29=1 Replace y with 25%2F3%2B%282%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax%2B1%2A%2825%2F3%29%2B1%282%2F3%29x=1 Distribute 1 to 25%2F3%2B%282%2F3%29%2Ax

3%2Ax%2B25%2F3%2B%282%2F3%29%2Ax=1 Multiply



3%2Ax%2B25%2F3%2B%282%2F3%29%2Ax=1 Reduce any fractions

3%2Ax%2B%282%2F3%29%2Ax=1-25%2F3 Subtract 25%2F3 from both sides


3%2Ax%2B%282%2F3%29%2Ax=3%2F3-25%2F3 Make 1 into a fraction with a denominator of 3


3%2Ax%2B%282%2F3%29%2Ax=-22%2F3 Combine the terms on the right side



%289%2F3%29%2Ax%2B%282%2F3%29x=-22%2F3 Make 3 into a fraction with a denominator of 3

%2811%2F3%29%2Ax=-22%2F3 Now combine the terms on the left side.


cross%28%283%2F11%29%2811%2F3%29%29x=%28-22%2F3%29%283%2F11%29 Multiply both sides by 3%2F11. This will cancel out 11%2F3 and isolate x

So when we multiply -22%2F3 and 3%2F11 (and simplify) we get



x=-2 <---------------------------------One answer

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

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

-6%2B1%2Ay=1 Multiply

1%2Ay=1%2B6Add 6 to both sides

1%2Ay=7 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%287%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=7%2F1 Multiply the terms on the right side


y=7 Reduce


So this is the other answer


y=7<---------------------------------Other answer


So our solution is

x=-2 and y=7

which can also look like

(-2,7)

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

2%2Ax-3%2Ay=-25
3%2Ax%2B1%2Ay=1

we get


graph of 2%2Ax-3%2Ay=-25 (red) and 3%2Ax%2B1%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 (-2,7). This verifies our answer.


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Check:

Plug in (-2,7) into the system of equations


Let x=-2 and y=7. Now plug those values into the equation 2%2Ax-3%2Ay=-25

2%2A%28-2%29-3%2A%287%29=-25 Plug in x=-2 and y=7


-4-21=-25 Multiply


-25=-25 Add


-25=-25 Reduce. Since this equation is true the solution works.


So the solution (-2,7) satisfies 2%2Ax-3%2Ay=-25



Let x=-2 and y=7. Now plug those values into the equation 3%2Ax%2B1%2Ay=1

3%2A%28-2%29%2B1%2A%287%29=1 Plug in x=-2 and y=7


-6%2B7=1 Multiply


1=1 Add


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


So the solution (-2,7) satisfies 3%2Ax%2B1%2Ay=1


Since the solution (-2,7) satisfies the system of equations


2%2Ax-3%2Ay=-25
3%2Ax%2B1%2Ay=1


this verifies our answer.