SOLUTION: Please help me solve this system of equations X=y-11 and x-3y=1

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Question 948990: Please help me solve this system of equations
X=y-11 and x-3y=1
Found 2 solutions by macston, MathLover1:
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
X=y-11 and x-3y=1
x=y-11
x-3y=1 Add 3y to each side
x=3y+1 Substitute first equation for x
y-11=3y+1 Subtract y from each side
-11=2y+1 Subtract 1 from each side
-12=2y Divide each side by 2
-6=y ANSWER 1:y=-6
x=y-11=-6-11=-17 ANSWER 2: x=-17
CHECK
x=y-11
-17=-6-11
-17=-17
x-3y=1
-17-3(-6)=1
-17-(-18)=1
1=1

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

x=y-11
x-3y=1
----------------
x-y=-11
x-3y=1

Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax-1%2Ay=-11
1%2Ax-3%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=-11-1%2AxSubtract 1%2Ax from both sides

y=%28-11-1%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=11%2B1%2Ax Now we've fully isolated y

Since y equals 11%2B1%2Ax we can substitute the expression 11%2B1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-3%2Ahighlight%28%2811%2B1%2Ax%29%29=1 Replace y with 11%2B1%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-3%2A%2811%29-3%281%29x=1 Distribute -3 to 11%2B1%2Ax

1%2Ax-33-3%2Ax=1 Multiply



1%2Ax-33-3%2Ax=1 Reduce any fractions

1%2Ax-3%2Ax=1%2B33Add 33 to both sides


1%2Ax-3%2Ax=34 Combine the terms on the right side



-2%2Ax=34 Now combine the terms on the left side.


cross%28%281%2F-2%29%28-2%2F1%29%29x=%2834%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 34%2F1 and 1%2F-2 (and simplify) we get



x=-17 <---------------------------------One answer

Now that we know that x=-17, lets substitute that in for x to solve for y

1%28-17%29-3%2Ay=1 Plug in x=-17 into the 2nd equation

-17-3%2Ay=1 Multiply

-3%2Ay=1%2B17Add 17 to both sides

-3%2Ay=18 Combine the terms on the right side

cross%28%281%2F-3%29%28-3%29%29%2Ay=%2818%2F1%29%281%2F-3%29 Multiply both sides by 1%2F-3. This will cancel out -3 on the left side.

y=18%2F-3 Multiply the terms on the right side


y=-6 Reduce


So this is the other answer


y=-6<---------------------------------Other answer


So our solution is

x=-17 and y=-6

which can also look like

(-17,-6)

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

1%2Ax-1%2Ay=-11
1%2Ax-3%2Ay=1

we get


graph of 1%2Ax-1%2Ay=-11 (red) and 1%2Ax-3%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 (-17,-6). This verifies our answer.


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

Plug in (-17,-6) into the system of equations


Let x=-17 and y=-6. Now plug those values into the equation 1%2Ax-1%2Ay=-11

1%2A%28-17%29-1%2A%28-6%29=-11 Plug in x=-17 and y=-6


-17%2B6=-11 Multiply


-11=-11 Add


-11=-11 Reduce. Since this equation is true the solution works.


So the solution (-17,-6) satisfies 1%2Ax-1%2Ay=-11



Let x=-17 and y=-6. Now plug those values into the equation 1%2Ax-3%2Ay=1

1%2A%28-17%29-3%2A%28-6%29=1 Plug in x=-17 and y=-6


-17%2B18=1 Multiply


1=1 Add


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


So the solution (-17,-6) satisfies 1%2Ax-3%2Ay=1


Since the solution (-17,-6) satisfies the system of equations


1%2Ax-1%2Ay=-11
1%2Ax-3%2Ay=1


this verifies our answer.