SOLUTION: I am having a problem understanding how to solve a system byusing substitution. Here are few of the problems I'm facing. Could you please explain them to me? 1. y = -3x+19

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: I am having a problem understanding how to solve a system byusing substitution. Here are few of the problems I'm facing. Could you please explain them to me? 1. y = -3x+19       Log On


   

Question 173267: I am having a problem understanding how to solve a system byusing substitution.
Here are few of the problems I'm facing. Could you please explain them to me?
1. y = -3x+19
y + 2x-1
2. y = x +4
3x-2y=6
3. 2x-y = 4
2x-y=3
I've looked at tons of reasons on how these problems work, and none of them make sense. The ones I've looked at are not specific enough on the what, why and how of the problem.
Thanks for your help!
Angela
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
The goal of solving ANY system of equations with 2 variables is to eliminate one variable so you can solve for the other variable. To eliminate one variable, simply "substitute" an expression in terms of the other variable.


# 1

Note: I'm assuming that the second equation is y+=+2x-1


y+=+-3x%2B19 Start with the first equation


2x-1+=+-3x%2B19 Plug in y+=+2x-1. In other words, replace "y" with 2x-1. Notice how the "y" term is gone.


Now we can solve for "x".


2x=-3x%2B19%2B1 Add 1 to both sides.


2x%2B3x=19%2B1 Add 3x to both sides.


5x=19%2B1 Combine like terms on the left side.


5x=20 Combine like terms on the right side.


x=%2820%29%2F%285%29 Divide both sides by 5 to isolate x.


x=4 Reduce. So this is the first answer.


--------------------------------


y+=+2x-1 Go back to the second equation


y+=+2%284%29-1 Plug in x=4


y+=+8-1 Multiply


y+=+7 Subtract. So this is the second answer.


=========================================

Answer:


So the solutions are x=4 and y+=+7


which form the ordered pair (4,7)


So the system is consistent and independent.





# 2


3x-2y=6+ Start with the second equation


3x-2%28x%2B4%29=6+ Plug in y+=+x+%2B4. In other words, replace "y" with x+4. Notice how the "y" term is gone.


Now we can solve for "x".


3x-2x-8=6 Distribute.


x-8=6 Combine like terms on the left side.


x=6%2B8 Add 8 to both sides.


x=14 Combine like terms on the right side. So this is the first answer.


-------------------------------------------


y+=+x+%2B4 Go back to the first equation


y+=+14+%2B4 Plug in x=14


y+=+18 Add. So this is the second answer.


=========================================

Answer:


So the solutions are x=14 and y+=+18


which form the ordered pair (14,18)


So the system is consistent and independent.




# 3





Start with the given system of equations:

system%282x-y=4%2C2x-y=3%29



Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation


2x-y=4 Start with the first equation


-y=4-2x Subtract 2x from both sides


-y=-2x%2B4 Rearrange the equation


y=%28-2x%2B4%29%2F%28-1%29 Divide both sides by -1


y=%28%28-2%29%2F%28-1%29%29x%2B%284%29%2F%28-1%29 Break up the fraction


y=2x-4 Reduce



---------------------

Since y=2x-4, we can now replace each y in the second equation with 2x-4 to solve for x



2x-highlight%28%282x-4%29%29=3 Plug in y=2x-4 into the second equation. In other words, replace each y with 2x-4. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.



2x-2x%2B4=3 Distribute the negative


4=3 Combine like terms on the left side


Since this equation is NEVER true for any x value, this means there are no solutions.


So the system is inconsistent.