SOLUTION: Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent. x+3y=2 x+y=1

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Question 177966: Solve each system by substitution. Determine whether the equations are independent, dependent, or inconsistent.
x+3y=2
x+y=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

1%2Ax%2B3%2Ay=2
1%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=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%2B1%2Ahighlight%28%282%2F3-%281%2F3%29%2Ax%29%29=1 Replace y with 2%2F3-%281%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B1%2A%282%2F3%29%2B1%28-1%2F3%29x=1 Distribute 1 to 2%2F3-%281%2F3%29%2Ax

1%2Ax%2B2%2F3-%281%2F3%29%2Ax=1 Multiply



1%2Ax%2B2%2F3-%281%2F3%29%2Ax=1 Reduce any fractions

1%2Ax-%281%2F3%29%2Ax=1-2%2F3 Subtract 2%2F3 from both sides


1%2Ax-%281%2F3%29%2Ax=3%2F3-2%2F3 Make 1 into a fraction with a denominator of 3


1%2Ax-%281%2F3%29%2Ax=1%2F3 Combine the terms on the right side



%283%2F3%29%2Ax-%281%2F3%29x=1%2F3 Make 1 into a fraction with a denominator of 3

%282%2F3%29%2Ax=1%2F3 Now combine the terms on the left side.


cross%28%283%2F2%29%282%2F3%29%29x=%281%2F3%29%283%2F2%29 Multiply both sides by 3%2F2. This will cancel out 2%2F3 and isolate x

So when we multiply 1%2F3 and 3%2F2 (and simplify) we get



x=1%2F2 <---------------------------------One answer

Now that we know that x=1%2F2, lets substitute that in for x to solve for y

1%281%2F2%29%2B1%2Ay=1 Plug in x=1%2F2 into the 2nd equation

1%2F2%2B1%2Ay=1 Multiply

1%2Ay=1-1%2F2Subtract 1%2F2 from both sides

1%2Ay=2%2F2-1%2F2 Make 1 into a fraction with a denominator of 2



1%2Ay=1%2F2 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%281%2F2%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=1%2F2 Multiply the terms on the right side


y=1%2F2 Reduce


So this is the other answer


y=1%2F2<---------------------------------Other answer


So our solution is

x=1%2F2 and y=1%2F2

which can also look like

(1%2F2,1%2F2)

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

1%2Ax%2B3%2Ay=2
1%2Ax%2B1%2Ay=1

we get


graph of 1%2Ax%2B3%2Ay=2 (red) and 1%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 (1%2F2,1%2F2). This verifies our answer.


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Check:

Plug in (1%2F2,1%2F2) into the system of equations


Let x=1%2F2 and y=1%2F2. Now plug those values into the equation 1%2Ax%2B3%2Ay=2

1%2A%281%2F2%29%2B3%2A%281%2F2%29=2 Plug in x=1%2F2 and y=1%2F2


1%2F2%2B3%2F2=2 Multiply


4%2F2=2 Add


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


So the solution (1%2F2,1%2F2) satisfies 1%2Ax%2B3%2Ay=2



Let x=1%2F2 and y=1%2F2. Now plug those values into the equation 1%2Ax%2B1%2Ay=1

1%2A%281%2F2%29%2B1%2A%281%2F2%29=1 Plug in x=1%2F2 and y=1%2F2


1%2F2%2B1%2F2=1 Multiply


2%2F2=1 Add


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


So the solution (1%2F2,1%2F2) satisfies 1%2Ax%2B1%2Ay=1


Since the solution (1%2F2,1%2F2) satisfies the system of equations


1%2Ax%2B3%2Ay=2
1%2Ax%2B1%2Ay=1


this verifies our answer.