SOLUTION: Hi I need help solving the system of equations by graphing y=-3x+3 5x+y=5 Please use method of substitution and elimination to solve (use both please). Please show me your st

Algebra ->  Linear-equations -> SOLUTION: Hi I need help solving the system of equations by graphing y=-3x+3 5x+y=5 Please use method of substitution and elimination to solve (use both please). Please show me your st      Log On


   

Question 145822: Hi I need help solving the system of equations by graphing
y=-3x+3
5x+y=5
Please use method of substitution and elimination to solve (use both please). Please show me your steps because I keep doing this wrong.
Thanks so much for your help
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Substitution:


Start with the given system
5x%2By=5
y=-3x%2B3



5x%2B%28-3x%2B3%29=5 Plug in y=-3x%2B3 into the first equation. In other words, replace each y with -3x%2B3. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.


2x%2B3=5 Combine like terms on the left side


2x=5-3Subtract 3 from both sides


2x=2 Combine like terms on the right side


x=%282%29%2F%282%29 Divide both sides by 2 to isolate x



x=1 Divide




Now that we know that x=1, we can plug this into y=-3x%2B3 to find y



y=-3%281%29%2B3 Substitute 1 for each x


y=0 Simplify


So our answer is x=1 and y=0 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%28+500%2C+500%2C+-5%2C+5%2C+-5%2C+5%2C+%285-5x%29%2F1%2C+-3x%2B3%29+ Graph of 5x%2By=5 (red) and y=-3x%2B3 (green)






Elimination:

y=-3x%2B3 Start with the first equation


3x%2By=3 Add 3x to both sides




So we now have the system of equations:

system%283x%2By=3%2C5x%2By=5%29



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





In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).


So lets eliminate x. In order to do that, we need to have both x coefficients that are equal in magnitude but have opposite signs (for instance 2 and -2 are equal in magnitude but have opposite signs). This way they will add to zero. By adding to zero, they can be eliminated.



So to make the x coefficients equal in magnitude but opposite in sign, we need to multiply both x coefficients by some number to get them to an common number. So if we wanted to get 3 and 5 to some equal number, we could try to get them to the LCM.



Since the LCM of 3 and 5 is 15, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by -3 like this:




5%283x%2By%29=5%283%29 Multiply the top equation (both sides) by 5
-3%285x%2By%29=-3%285%29 Multiply the bottom equation (both sides) by -3




Distribute and multiply

15x%2B5y=15
-15x-3y=-15


Now add the equations together. In order to add 2 equations, group like terms and combine them

%2815x-15x%29%2B%285y-3y%29=15-15

Combine like terms and simplify



cross%2815x-15x%29%2B2y=0 Notice how the x terms cancel out




2y=0 Simplify




y=0%2F2 Divide both sides by 2 to isolate y




y=0 Reduce



Now plug this answer into the top equation 3x%2By=3 to solve for x

3x%2By=3 Start with the first equation



3x%2B%280%29=3 Plug in y=0




3x%2B0=3 Multiply



3x=3-0Subtract 0 from both sides


3x=3 Combine like terms on the right side


x=%283%29%2F%283%29 Divide both sides by 3 to isolate x



x=1 Divide




So our answer is
x=1 and y=0



which also looks like




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 3x%2By=3 (red) and 5x%2By=5 (green) and the intersection of the lines (blue circle).