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

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> 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       Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
#1


Start with the given system
y=2x%2B6
y=-x-3



%28-x-3%29=2x%2B6 Plug in y=-x-3 into the first equation. In other words, replace each y with -x-3. Notice we've eliminated the y variables. So we now have a simple equation with one unknown.


%28-x-3%29=2x%2B6 Distribute


-x-3=2x%2B6 Combine like terms on the left side


-x=2x%2B6%2B3Add 3 to both sides


-x-2x=6%2B3 Subtract 2x from both sides


-3x=6%2B3 Combine like terms on the left side


-3x=9 Combine like terms on the right side


x=%289%29%2F%28-3%29 Divide both sides by -3 to isolate x



x=-3 Divide




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



y=-%28-3%29-3 Substitute -3 for each x


y=0 Simplify


So our answer is x=-3 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+2x%2B6%2C+-x-3%29+ Graph of y=2x%2B6 (red) and y=-x-3 (green)





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


Lets start with the given system of linear equations

-3%2Ax%2B1%2Ay=9
1%2Ax%2B2%2Ay=4

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

1%2Ay=9%2B3%2AxAdd 3%2Ax to both sides

y=%289%2B3%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=9%2B3%2Ax Now we've fully isolated y

Since y equals 9%2B3%2Ax we can substitute the expression 9%2B3%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B2%2Ahighlight%28%289%2B3%2Ax%29%29=4 Replace y with 9%2B3%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B2%2A%289%29%2B2%283%29x=4 Distribute 2 to 9%2B3%2Ax

1%2Ax%2B18%2B6%2Ax=4 Multiply



1%2Ax%2B18%2B6%2Ax=4 Reduce any fractions

1%2Ax%2B6%2Ax=4-18 Subtract 18 from both sides


1%2Ax%2B6%2Ax=-14 Combine the terms on the right side



7%2Ax=-14 Now combine the terms on the left side.


cross%28%281%2F7%29%287%2F1%29%29x=%28-14%2F1%29%281%2F7%29 Multiply both sides by 1%2F7. This will cancel out 7%2F1 and isolate x

So when we multiply -14%2F1 and 1%2F7 (and simplify) we get



x=-2 <---------------------------------One answer

Now that we know that x=-2, lets substitute that in for x to solve for y

1%28-2%29%2B2%2Ay=4 Plug in x=-2 into the 2nd equation

-2%2B2%2Ay=4 Multiply

2%2Ay=4%2B2Add 2 to both sides

2%2Ay=6 Combine the terms on the right side

cross%28%281%2F2%29%282%29%29%2Ay=%286%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.

y=6%2F2 Multiply the terms on the right side


y=3 Reduce


So this is the other answer


y=3<---------------------------------Other answer


So our solution is

x=-2 and y=3

which can also look like

(-2,3)

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

-3%2Ax%2B1%2Ay=9
1%2Ax%2B2%2Ay=4

we get


graph of -3%2Ax%2B1%2Ay=9 (red) and 1%2Ax%2B2%2Ay=4 (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 (-2,3). This verifies our answer.


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

Plug in (-2,3) into the system of equations


Let x=-2 and y=3. Now plug those values into the equation -3%2Ax%2B1%2Ay=9

-3%2A%28-2%29%2B1%2A%283%29=9 Plug in x=-2 and y=3


6%2B3=9 Multiply


9=9 Add


9=9 Reduce. Since this equation is true the solution works.


So the solution (-2,3) satisfies -3%2Ax%2B1%2Ay=9



Let x=-2 and y=3. Now plug those values into the equation 1%2Ax%2B2%2Ay=4

1%2A%28-2%29%2B2%2A%283%29=4 Plug in x=-2 and y=3


-2%2B6=4 Multiply


4=4 Add


4=4 Reduce. Since this equation is true the solution works.


So the solution (-2,3) satisfies 1%2Ax%2B2%2Ay=4


Since the solution (-2,3) satisfies the system of equations


-3%2Ax%2B1%2Ay=9
1%2Ax%2B2%2Ay=4


this verifies our answer.









#3

y+%2B+4+=+2x Start with the first equation


y=+2x-4 Solve for y by subtracting 4 from both sides




Start with the given system
6x-3y=12
y=2x-4



6x-3%282x-4%29=12 Plug in y=2x-4 into the first 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.


6x-6x%2B12=12 Distribute


12=12 Combine like terms on the left side


0=12-12Subtract 12 from both sides


0=0 Combine like terms on the right side


0=0 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.