SOLUTION: solve the system of equations using the substitution mehtod 3x+y=2 2x-y=3

Algebra ->  Graphs -> SOLUTION: solve the system of equations using the substitution mehtod 3x+y=2 2x-y=3      Log On


   

Question 120364: solve the system of equations using the substitution mehtod
3x+y=2
2x-y=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

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

1%2Ay=2-3%2AxSubtract 3%2Ax from both sides

y=%282-3%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=2-3%2Ax Now we've fully isolated y

Since y equals 2-3%2Ax we can substitute the expression 2-3%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


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

2%2Ax-1%2A%282%29-1%28-3%29x=3 Distribute -1 to 2-3%2Ax

2%2Ax-2%2B3%2Ax=3 Multiply



2%2Ax-2%2B3%2Ax=3 Reduce any fractions

2%2Ax%2B3%2Ax=3%2B2Add 2 to both sides


2%2Ax%2B3%2Ax=5 Combine the terms on the right side



5%2Ax=5 Now combine the terms on the left side.


cross%28%281%2F5%29%285%2F1%29%29x=%285%2F1%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5%2F1 and isolate x

So when we multiply 5%2F1 and 1%2F5 (and simplify) we get



x=1 <---------------------------------One answer

Now that we know that x=1, lets substitute that in for x to solve for y

2%281%29-1%2Ay=3 Plug in x=1 into the 2nd equation

2-1%2Ay=3 Multiply

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

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

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

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


y=-1 Reduce


So this is the other answer


y=-1<---------------------------------Other answer


So our solution is

x=1 and y=-1

which can also look like

(1,-1)

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

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

we get


graph of 3%2Ax%2B1%2Ay=2 (red) and 2%2Ax-1%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,-1). This verifies our answer.


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

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


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

3%2A%281%29%2B1%2A%28-1%29=2 Plug in x=1 and y=-1


3-1=2 Multiply


2=2 Add


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


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



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

2%2A%281%29-1%2A%28-1%29=3 Plug in x=1 and y=-1


2%2B1=3 Multiply


3=3 Add


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


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


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


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


this verifies our answer.