SOLUTION: 2x + y = 4 3x - 2y = -1

Algebra ->  Linear-equations -> SOLUTION: 2x + y = 4 3x - 2y = -1      Log On


   

Question 117993This question is from textbook
: 2x + y = 4
3x - 2y = -1 This question is from textbook

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%2B1%2Ay=4
3%2Ax-2%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

1%2Ay=4-2%2AxSubtract 2%2Ax from both sides

y=%284-2%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=4-2%2Ax Now we've fully isolated y

Since y equals 4-2%2Ax we can substitute the expression 4-2%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B-2%2Ahighlight%28%284-2%2Ax%29%29=-1 Replace y with 4-2%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax-2%2A%284%29-2%28-2%29x=-1 Distribute -2 to 4-2%2Ax

3%2Ax-8%2B4%2Ax=-1 Multiply



3%2Ax-8%2B4%2Ax=-1 Reduce any fractions

3%2Ax%2B4%2Ax=-1%2B8Add 8 to both sides


3%2Ax%2B4%2Ax=7 Combine the terms on the right side



7%2Ax=7 Now combine the terms on the left side.


cross%28%281%2F7%29%287%2F1%29%29x=%287%2F1%29%281%2F7%29 Multiply both sides by 1%2F7. This will cancel out 7%2F1 and isolate x

So when we multiply 7%2F1 and 1%2F7 (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

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

3-2%2Ay=-1 Multiply

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

-2%2Ay=-4 Combine the terms on the right side

cross%28%281%2F-2%29%28-2%29%29%2Ay=%28-4%2F1%29%281%2F-2%29 Multiply both sides by 1%2F-2. This will cancel out -2 on the left side.

y=-4%2F-2 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 and y=2

which can also look like

(1,2)

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

2%2Ax%2B1%2Ay=4
3%2Ax-2%2Ay=-1

we get


graph of 2%2Ax%2B1%2Ay=4 (red) and 3%2Ax-2%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 (1,2). This verifies our answer.


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

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


Let x=1 and y=2. Now plug those values into the equation 2%2Ax%2B1%2Ay=4

2%2A%281%29%2B1%2A%282%29=4 Plug in x=1 and y=2


2%2B2=4 Multiply


4=4 Add


4=4 Reduce. Since this equation is true the solution works.


So the solution (1,2) satisfies 2%2Ax%2B1%2Ay=4



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

3%2A%281%29-2%2A%282%29=-1 Plug in x=1 and y=2


3-4=-1 Multiply


-1=-1 Add


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


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


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


2%2Ax%2B1%2Ay=4
3%2Ax-2%2Ay=-1


this verifies our answer.