SOLUTION: I need help with the following problems: y = 2x + 6 y = -x -3 y - 3x = 9 2y + x = 4 y + 4 = 2x 6x - 3y = 12

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Question 121939: I need help with the following problems:
y = 2x + 6
y = -x -3

y - 3x = 9
2y + x = 4


y + 4 = 2x
6x - 3y = 12




Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
#1


Start with the given system





Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.


Distribute


Combine like terms on the left side


Add 3 to both sides


Subtract 2x from both sides


Combine like terms on the left side


Combine like terms on the right side


Divide both sides by -3 to isolate x



Divide




Now that we know that , we can plug this into to find



Substitute for each


Simplify


So our answer is and which also looks like



Notice if we graph the two equations, we can see that their intersection is at . So this verifies our answer.


Graph of (red) and (green)





#2
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations




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

Add to both sides

Divide both sides by 1.


Which breaks down and reduces to



Now we've fully isolated y

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


Replace y with . Since this eliminates y, we can now solve for x.

Distribute 2 to

Multiply



Reduce any fractions

Subtract from both sides


Combine the terms on the right side



Now combine the terms on the left side.


Multiply both sides by . This will cancel out and isolate x

So when we multiply and (and simplify) we get



<---------------------------------One answer

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

Plug in into the 2nd equation

Multiply

Add to both sides

Combine the terms on the right side

Multiply both sides by . This will cancel out 2 on the left side.

Multiply the terms on the right side


Reduce


So this is the other answer


<---------------------------------Other answer


So our solution is

and

which can also look like

(,)

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




we get


graph of (red) and (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 (,). This verifies our answer.


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

Plug in (,) into the system of equations


Let and . Now plug those values into the equation

Plug in and


Multiply


Add


Reduce. Since this equation is true the solution works.


So the solution (,) satisfies



Let and . Now plug those values into the equation

Plug in and


Multiply


Add


Reduce. Since this equation is true the solution works.


So the solution (,) satisfies


Since the solution (,) satisfies the system of equations






this verifies our answer.









#3

Start with the first equation


Solve for y by subtracting 4 from both sides




Start with the given system





Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.


Distribute


Combine like terms on the left side


Subtract 12 from both sides


Combine like terms on the right side


Simplify

Since this equation is always true for any x value, this means x can equal any number. So there are an infinite number of solutions.
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