SOLUTION: I have not been able to figure out how to do the following problem. x + 3y =2 x - 2y =-3

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Question 99569This question is from textbook Fundamentals of Algebraic Modeling
: I have not been able to figure out how to do the following problem.
x + 3y =2
x - 2y =-3 This question is from textbook Fundamentals of Algebraic Modeling

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

1%2Ax%2B3%2Ay=2
1%2Ax-2%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

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

y=%282-1%2Ax%29%2F3 Divide both sides by 3.


Which breaks down and reduces to



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

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


1%2Ax%2B-2%2Ahighlight%28%282%2F3-%281%2F3%29%2Ax%29%29=-3 Replace y with 2%2F3-%281%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-2%2A%282%2F3%29-2%28-1%2F3%29x=-3 Distribute -2 to 2%2F3-%281%2F3%29%2Ax

1%2Ax-4%2F3%2B%282%2F3%29%2Ax=-3 Multiply



1%2Ax-4%2F3%2B%282%2F3%29%2Ax=-3 Reduce any fractions

1%2Ax%2B%282%2F3%29%2Ax=-3%2B4%2F3Add 4%2F3 to both sides


1%2Ax%2B%282%2F3%29%2Ax=-9%2F3%2B4%2F3 Make -3 into a fraction with a denominator of 3


1%2Ax%2B%282%2F3%29%2Ax=-5%2F3 Combine the terms on the right side



%283%2F3%29%2Ax%2B%282%2F3%29x=-5%2F3 Make 1 into a fraction with a denominator of 3

%285%2F3%29%2Ax=-5%2F3 Now combine the terms on the left side.


cross%28%283%2F5%29%285%2F3%29%29x=%28-5%2F3%29%283%2F5%29 Multiply both sides by 3%2F5. This will cancel out 5%2F3 and isolate x

So when we multiply -5%2F3 and 3%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

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

-1-2%2Ay=-3 Multiply

-2%2Ay=-3%2B1Add 1 to both sides

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

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

y=-2%2F-2 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)

1%2Ax%2B3%2Ay=2
1%2Ax-2%2Ay=-3

we get


graph of 1%2Ax%2B3%2Ay=2 (red) and 1%2Ax-2%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.


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Check:

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


Let x=-1 and y=1. Now plug those values into the equation 1%2Ax%2B3%2Ay=2

1%2A%28-1%29%2B3%2A%281%29=2 Plug in x=-1 and y=1


-1%2B3=2 Multiply


2=2 Add


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


So the solution (-1,1) satisfies 1%2Ax%2B3%2Ay=2



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

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


-1-2=-3 Multiply


-3=-3 Add


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


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


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


1%2Ax%2B3%2Ay=2
1%2Ax-2%2Ay=-3


this verifies our answer.