SOLUTION: Solve for X and Y in the following problems. Make sure you show all your work so you can get partial credit even if your final answer is wrong. a. X + Y=6 , 2X + Y = 8

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: Solve for X and Y in the following problems. Make sure you show all your work so you can get partial credit even if your final answer is wrong. a. X + Y=6 , 2X + Y = 8       Log On


   

Question 174547: Solve for X and Y in the following problems. Make sure you show all your work so you can get partial credit even if your final answer is wrong.
a. X + Y=6 , 2X + Y = 8
b. 7X + 3Y = 14 , 5X + 9Y = 10
c. 4X + Y = 16 , 2X + 3Y = 24
d. 12X + Y = 25 , 8X - 2Y = 14
I get the variables and graphing, however have not found any examples of how to solve only given this amount of information.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first two to get you started


a)


Start with the given system of equations:

system%28x%2By=6%2C2x%2By=8%29


Let's solve the system by substitution


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

x%2By=6 Start with the first equation


y=6-x Subtract x from both sides


y=-x%2B6 Rearrange the equation


y=%28-x%2B6%29%2F%281%29 Divide both sides by 1


y=%28%28-1%29%2F%281%29%29x%2B%286%29%2F%281%29 Break up the fraction


y=-x%2B6 Reduce



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

Since y=-x%2B6, we can now replace each y in the second equation with -x%2B6 to solve for x



2x%2Bhighlight%28%28-x%2B6%29%29=8 Plug in y=-x%2B6 into the second equation. In other words, replace each y with -x%2B6. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.



x%2B6=8 Combine like terms on the left side


x=8-6Subtract 6 from both sides


x=2 Combine like terms on the right side





-----------------First Answer------------------------------


So the first part of our answer is: x=2









Since we know that x=2 we can plug it into the equation y=-x%2B6 (remember we previously solved for y in the first equation).



y=-x%2B6 Start with the equation where y was previously isolated.


y=-%282%29%2B6 Plug in x=2


y=-2%2B6 Multiply


y=4 Combine like terms



-----------------Second Answer------------------------------


So the second part of our answer is: y=4









-----------------Summary------------------------------

So our answers are:

x=2 and y=4

which form the point








Now let's graph the two equations (if you need help with graphing, check out this solver)


From the graph, we can see that the two equations intersect at . This visually verifies our answer.




graph of x%2By=6 (red) and 2x%2By=8 (green) and the intersection of the lines (blue circle).







b)


Start with the given system of equations:
system%287x%2B3y=14%2C5x%2B9y=10%29


-3%287x%2B3y%29=-3%2814%29 Multiply the both sides of the first equation by -3.


-21x-9y=-42 Distribute and multiply.


So we have the new system of equations:
system%28-21x-9y=-42%2C5x%2B9y=10%29


Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:


%28-21x-9y%29%2B%285x%2B9y%29=%28-42%29%2B%2810%29


%28-21x%2B5x%29%2B%28-9y%2B9y%29=-42%2B10 Group like terms.


-16x%2B0y=-32 Combine like terms. Notice how the y terms cancel out.


-16x=-32 Simplify.


x=%28-32%29%2F%28-16%29 Divide both sides by -16 to isolate x.


x=2 Reduce.


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


-21x-9y=-42 Now go back to the first equation.


-21%282%29-9y=-42 Plug in x=2.


-42-9y=-42 Multiply.


-9y=-42%2B42 Add 42 to both sides.


-9y=0 Combine like terms on the right side.


y=%280%29%2F%28-9%29 Divide both sides by -9 to isolate y.


y=0 Reduce.


So our answer is x=2 and y=0.


Which form the ordered pair .


This means that the system is consistent and independent.


Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer.


Graph of 7x%2B3y=14 (red) and 5x%2B9y=10 (green)