SOLUTION: Linear system addition and subtraction im having trouble showing my work to get these answers question 1 answer = (-1,3) x/3 + y/2 = 1/6 x - 6y = 8

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Question 82846: Linear system addition and subtraction
im having trouble showing my work to get these answers
question 1 answer = (-1,3)
x/3 + y/2 = 1/6
x - 6y = 8

question 2 answer = (2,-1)
3x + 5y = 12
7x + 5y = 8
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Start with the given system of equations
x%2F3+%2B+y%2F2+=+1%2F6
x+-+6y+=+8+

Multiply the top equation by 6 to get integer coefficients (this eliminates the denominator)
6%28x%2F3+%2B+y%2F2+%29=+6%281%2F6%29 Multiply both sides by 6
2x%2B3y=+1 Distribute

So now the system becomes


2x%2B3y=+1
x+-+6y+=+8+

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

2%2Ax%2B3%2Ay=1
1%2Ax-6%2Ay=8

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 but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 2 and 1 to some equal number, we could try to get them to the LCM.

Since the LCM of 2 and 1 is 2, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -2 like this:

1%2A%282%2Ax%2B3%2Ay%29=%281%29%2A1 Multiply the top equation (both sides) by 1
-2%2A%281%2Ax-6%2Ay%29=%288%29%2A-2 Multiply the bottom equation (both sides) by -2


So after multiplying we get this:
2%2Ax%2B3%2Ay=1
-2%2Ax%2B12%2Ay=-16

Notice how 2 and -2 add to zero (ie 2%2B-2=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%282%2Ax-2%2Ax%29%2B%283%2Ay%2B12%2Ay%29=1-16

%282-2%29%2Ax%2B%283%2B12%29y=1-16

cross%282%2B-2%29%2Ax%2B%283%2B12%29%2Ay=1-16 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

15%2Ay=-15

y=-15%2F15 Divide both sides by 15 to solve for y



y=-1 Reduce


Now plug this answer into the top equation 2%2Ax%2B3%2Ay=1 to solve for x

2%2Ax%2B3%28-1%29=1 Plug in y=-1


2%2Ax-3=1 Multiply



2%2Ax=1%2B3 Subtract -3 from both sides

2%2Ax=4 Combine the terms on the right side

cross%28%281%2F2%29%282%29%29%2Ax=%284%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.


x=2 Multiply the terms on the right side


So our answer is

x=2, y=-1

which also looks like

(2, -1)

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

2%2Ax%2B3%2Ay=1
1%2Ax-6%2Ay=8

we get



graph of 2%2Ax%2B3%2Ay=1 (red) 1%2Ax-6%2Ay=8 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (2,-1). This verifies our answer.


Check:
Plug in (2,-1) into the system
x%2F3+%2B+y%2F2+=+1%2F6
x+-+6y+=+8+
2%2F3+%2B+-1%2F2+=+1%2F6 Plug in x=2, y=-1
2+-+6%28-1%29+=+8+
4%2F6+-+3%2F6+=+1%2F6
2+%2B+6+=+8+
1%2F6+=+1%2F6 works
8=+8+ works
this verifies our answer

-----------------------------------------------------------------
Now lets look at

3x+%2B+5y+=+12
7x+%2B+5y+=+8

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

3%2Ax%2B5%2Ay=12
7%2Ax%2B5%2Ay=8

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 but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 3 and 7 to some equal number, we could try to get them to the LCM.

Since the LCM of 3 and 7 is 21, we need to multiply both sides of the top equation by 7 and multiply both sides of the bottom equation by -3 like this:

7%2A%283%2Ax%2B5%2Ay%29=%2812%29%2A7 Multiply the top equation (both sides) by 7
-3%2A%287%2Ax%2B5%2Ay%29=%288%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
21%2Ax%2B35%2Ay=84
-21%2Ax-15%2Ay=-24

Notice how 21 and -21 add to zero (ie 21%2B-21=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2821%2Ax-21%2Ax%29%2B%2835%2Ay-15%2Ay%29=84-24

%2821-21%29%2Ax%2B%2835-15%29y=84-24

cross%2821%2B-21%29%2Ax%2B%2835-15%29%2Ay=84-24 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

20%2Ay=60

y=60%2F20 Divide both sides by 20 to solve for y



y=3 Reduce


Now plug this answer into the top equation 3%2Ax%2B5%2Ay=12 to solve for x

3%2Ax%2B5%283%29=12 Plug in y=3


3%2Ax%2B15=12 Multiply



3%2Ax=12-15 Subtract 15 from both sides

3%2Ax=-3 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%28-3%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=-1 Multiply the terms on the right side


So our answer is

x=-1, y=3

which also looks like

(-1, 3)

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

3%2Ax%2B5%2Ay=12
7%2Ax%2B5%2Ay=8

we get



graph of 3%2Ax%2B5%2Ay=12 (red) 7%2Ax%2B5%2Ay=8 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (-1,3). This verifies our answer.


Check:
Plug in (-1,3) into the system
3x+%2B+5y+=+12
7x+%2B+5y+=+8
3%28-1%29+%2B+5%283%29+=+12 Plug in x=2, y=-1
7%28-1%29+%2B+5%283%29+=+8
-3+%2B15+=+12
-7+%2B15+=+8+
12+=+12 works
8=+8+ works
this verifies our answer