SOLUTION: I need help with four problems. You have to solve each system of equations by substitutions. Te first one i need help with is x+y=1 2x-y=-2. The second one is 5x-3y=-11 and x-2y=2

Algebra ->  Expressions-with-variables -> SOLUTION: I need help with four problems. You have to solve each system of equations by substitutions. Te first one i need help with is x+y=1 2x-y=-2. The second one is 5x-3y=-11 and x-2y=2      Log On


   

Question 127865: I need help with four problems. You have to solve each system of equations by substitutions. Te first one i need help with is x+y=1
2x-y=-2. The second one is 5x-3y=-11 and x-2y=2. The third one is x-y=3 and 6x+4y=13. The fourth one is 2x-y=16 and -x+2y=-8. I have no clue how to do these problems so i need help.
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


# 1




Start with the given system of equations:

system%28x%2By=1%2C2x-y=-2%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

x%2By=1 Start with the first equation


y=1-x Subtract x from both sides


y=-x%2B1 Rearrange the equation


y=%28-x%2B1%29%2F%281%29 Divide both sides by 1


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


y=-x%2B1 Reduce



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

Since y=-x%2B1, we can now replace each y in the second equation with -x%2B1 to solve for x



2x-highlight%28%28-x%2B1%29%29=-2 Plug in y=-x%2B1 into the first equation. In other words, replace each y with -x%2B1. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.



2x%2Bx-1=-2 Distribute the negative


3x-1=-2 Combine like terms on the left side


3x=-2%2B1Add 1 to both sides


3x=-1 Combine like terms on the right side


x=%28-1%29%2F%283%29 Divide both sides by 3 to isolate x



x=-1%2F3 Reduce





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


So the first part of our answer is: x=-1%2F3









Since we know that x=-1%2F3 we can plug it into the equation y=-x%2B1 (remember we previously solved for y in the first equation).



y=-x%2B1 Start with the equation where y was previously isolated.


y=-%28-1%2F3%29%2B1 Plug in x=-1%2F3


y=1%2F3%2B1 Multiply


y=4%2F3 Combine like terms (note: if you need help with fractions, check out this solver)



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


So the second part of our answer is: y=4%2F3









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

So our answers are:

x=-1%2F3 and y=4%2F3

which form the point








# 2





Start with the given system of equations:

system%285x-3y=-11%2Cx-2y=2%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

5x-3y=-11 Start with the first equation


-3y=-11-5x Subtract 5x from both sides


-3y=-5x-11 Rearrange the equation


y=%28-5x-11%29%2F%28-3%29 Divide both sides by -3


y=%28%28-5%29%2F%28-3%29%29x%2B%28-11%29%2F%28-3%29 Break up the fraction


y=%285%2F3%29x%2B11%2F3 Reduce



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

Since y=%285%2F3%29x%2B11%2F3, we can now replace each y in the second equation with %285%2F3%29x%2B11%2F3 to solve for x



x-2highlight%28%28%285%2F3%29x%2B11%2F3%29%29=2 Plug in y=%285%2F3%29x%2B11%2F3 into the first equation. In other words, replace each y with %285%2F3%29x%2B11%2F3. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.



x%2B%28-2%29%285%2F3%29x%2B%28-2%29%2811%2F3%29=2 Distribute -2 to %285%2F3%29x%2B11%2F3


x-%2810%2F3%29x-22%2F3=2 Multiply


%283%29%281x-%2810%2F3%29x-22%2F3%29=%283%29%282%29 Multiply both sides by the LCM of 3. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)



3x-10x-22=6 Distribute and multiply the LCM to each side



-7x-22=6 Combine like terms on the left side


-7x=6%2B22Add 22 to both sides


-7x=28 Combine like terms on the right side


x=%2828%29%2F%28-7%29 Divide both sides by -7 to isolate x



x=-4 Divide





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


So the first part of our answer is: x=-4









Since we know that x=-4 we can plug it into the equation y=%285%2F3%29x%2B11%2F3 (remember we previously solved for y in the first equation).



y=%285%2F3%29x%2B11%2F3 Start with the equation where y was previously isolated.


y=%285%2F3%29%28-4%29%2B11%2F3 Plug in x=-4


y=-20%2F3%2B11%2F3 Multiply


y=-3 Combine like terms and reduce. (note: if you need help with fractions, check out this solver)



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


So the second part of our answer is: y=-3









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

So our answers are:

x=-4 and y=-3

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 5x-3y=-11 (red) and x-2y=2 (green) and the intersection of the lines (blue circle).