SOLUTION: 3x+y=0 x+y=2 I'm not exactly sure on what catergory this falls under, but I don't understand the problem and basically what I'm doing.

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: 3x+y=0 x+y=2 I'm not exactly sure on what catergory this falls under, but I don't understand the problem and basically what I'm doing.      Log On


   

Question 82177: 3x+y=0
x+y=2

I'm not exactly sure on what catergory this falls under, but I don't understand the problem and basically what I'm doing.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
If you want to solve this system by substitution then...

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=0
1%2Ax%2B1%2Ay=2

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=0-3%2AxSubtract 3%2Ax from both sides

y=%280-3%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=0-3%2Ax Now we've fully isolated y

Since y equals 0-3%2Ax we can substitute the expression 0-3%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B1%2Ahighlight%28%280-3%2Ax%29%29=2 Replace y with 0-3%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B1%2A%280%29%2B1%28-3%29x=2 Distribute 1 to 0-3%2Ax

1%2Ax%2B0-3%2Ax=2 Multiply



1%2Ax%2B0-3%2Ax=2 Reduce any fractions

1%2Ax-3%2Ax=2%2B0Add 0 to both sides


1%2Ax-3%2Ax=2 Combine the terms on the right side



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


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

So when we multiply 2%2F1 and 1%2F-2 (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%2B1%2Ay=2 Plug in x=-1 into the 2nd equation

-1%2B1%2Ay=2 Multiply

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

1%2Ay=3 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%283%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=3%2F1 Multiply the terms on the right side


y=3 Reduce


So this is the other answer


y=3<---------------------------------Other answer


So our solution is

x=-1 and y=3

which can also look like

(-1,3)

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

3%2Ax%2B1%2Ay=0
1%2Ax%2B1%2Ay=2

we get


graph of 3%2Ax%2B1%2Ay=0 (red) and 1%2Ax%2B1%2Ay=2 (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 (-1,3). This verifies our answer.


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Check:

Plug in (-1,3) into the system of equations


Let x=-1 and y=3. Now plug those values into the equation 3%2Ax%2B1%2Ay=0

3%2A%28-1%29%2B1%2A%283%29=0 Plug in x=-1 and y=3


-3%2B3=0 Multiply


0=0 Add


0=0 Reduce. Since this equation is true the solution works.


So the solution (-1,3) satisfies 3%2Ax%2B1%2Ay=0



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

1%2A%28-1%29%2B1%2A%283%29=2 Plug in x=-1 and y=3


-1%2B3=2 Multiply


2=2 Add


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


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


Since the solution (-1,3) satisfies the system of equations


3%2Ax%2B1%2Ay=0
1%2Ax%2B1%2Ay=2


this verifies our answer.





Or if you want to solve by elimination:

Solved by pluggable solver: Linear Systems by Addition
We'll solve the system:
3%2Ax+%2B+1%2Ay+=+0
1%2Ax+%2B+1%2Ay+=+2
by elimination by addition.To eliminate by addition, we need to set both coefficients of x to numbers with changed signs, i.e a and -a. Since in the second equation we have 1 as our coefficient for x, to get -3 we have to multiply all terms of the second equation by -3%2F1 which is equal to -3.

Multiplying, we get on our second equation:%281%2A-3%29x+%2B+%281%2A-3%29y+=+2%2A-3
-3%2Ax+%2B+-3%2Ay+=+-6

Adding both equations we get:

%283%2B-3%29x+%2B+%281%2B-3%29y+=+%280%2B-6%29

Since 3 and -3 cancel out, we have a linear equation:Therefore, we know that y = 3.

Plugging that in into the first equation gives us:

3%2Ax+%2B+1%2Ay+=+0
3%2Ax+%2B+1%2A3+=+0
3%2Ax+%2B+3+=+0
3%2Ax+=+0+-+3
x+=+%280+-+3%29%2F3
x+=+-3%2F3
x+=+-1

Therefore, our answer is:

system%28+x=-1%2C+y=3+%29



Whatever method you choose, you'll get the same answer (provided you did it right of course).