SOLUTION: solve {3x-3y=1 {x+y=4

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Question 118321: solve {3x-3y=1
{x+y=4
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-3%2Ay=1
1%2Ax%2B1%2Ay=4

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

y=%281-3%2Ax%29%2F-3 Divide both sides by -3.


Which breaks down and reduces to



y=-1%2F3%2B1%2Ax Now we've fully isolated y

Since y equals -1%2F3%2B1%2Ax we can substitute the expression -1%2F3%2B1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B1%2Ahighlight%28%28-1%2F3%2B1%2Ax%29%29=4 Replace y with -1%2F3%2B1%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B1%2A%28-1%2F3%29%2B1%281%29x=4 Distribute 1 to -1%2F3%2B1%2Ax

1%2Ax-1%2F3%2B1%2Ax=4 Multiply



1%2Ax-1%2F3%2B1%2Ax=4 Reduce any fractions

1%2Ax%2B1%2Ax=4%2B1%2F3Add 1%2F3 to both sides


1%2Ax%2B1%2Ax=12%2F3%2B1%2F3 Make 4 into a fraction with a denominator of 3


1%2Ax%2B1%2Ax=13%2F3 Combine the terms on the right side



2%2Ax=13%2F3 Now combine the terms on the left side.


cross%28%281%2F2%29%282%2F1%29%29x=%2813%2F3%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2%2F1 and isolate x

So when we multiply 13%2F3 and 1%2F2 (and simplify) we get



x=13%2F6 <---------------------------------One answer

Now that we know that x=13%2F6, lets substitute that in for x to solve for y

1%2813%2F6%29%2B1%2Ay=4 Plug in x=13%2F6 into the 2nd equation

13%2F6%2B1%2Ay=4 Multiply

1%2Ay=4-13%2F6Subtract 13%2F6 from both sides

1%2Ay=24%2F6-13%2F6 Make 4 into a fraction with a denominator of 6



1%2Ay=11%2F6 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%2811%2F6%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=11%2F6 Multiply the terms on the right side


y=11%2F6 Reduce


So this is the other answer


y=11%2F6<---------------------------------Other answer


So our solution is

x=13%2F6 and y=11%2F6

which can also look like

(13%2F6,11%2F6)

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

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

we get


graph of 3%2Ax-3%2Ay=1 (red) and 1%2Ax%2B1%2Ay=4 (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 (13%2F6,11%2F6). This verifies our answer.


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Check:

Plug in (13%2F6,11%2F6) into the system of equations


Let x=13%2F6 and y=11%2F6. Now plug those values into the equation 3%2Ax-3%2Ay=1

3%2A%2813%2F6%29-3%2A%2811%2F6%29=1 Plug in x=13%2F6 and y=11%2F6


39%2F6-33%2F6=1 Multiply


6%2F6=1 Add


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


So the solution (13%2F6,11%2F6) satisfies 3%2Ax-3%2Ay=1



Let x=13%2F6 and y=11%2F6. Now plug those values into the equation 1%2Ax%2B1%2Ay=4

1%2A%2813%2F6%29%2B1%2A%2811%2F6%29=4 Plug in x=13%2F6 and y=11%2F6


13%2F6%2B11%2F6=4 Multiply


24%2F6=4 Add


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


So the solution (13%2F6,11%2F6) satisfies 1%2Ax%2B1%2Ay=4


Since the solution (13%2F6,11%2F6) satisfies the system of equations


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


this verifies our answer.