SOLUTION: Systems of Linear Equations help please?!?!? 16: 3x - 2y = 12 7x + 2y = 8 (3, 2) (2, -3) (-2, 1) 17: 3x + 5y = 15 6x + 10y =

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: Systems of Linear Equations help please?!?!? 16: 3x - 2y = 12 7x + 2y = 8 (3, 2) (2, -3) (-2, 1) 17: 3x + 5y = 15 6x + 10y =      Log On


   

Question 84868: Systems of Linear Equations help please?!?!?
16:
3x - 2y = 12
7x + 2y = 8
(3, 2)
(2, -3)
(-2, 1)

17:
3x + 5y = 15
6x + 10y = -30
(1, -1)
an infinite number of solutions
no solution

18:
y = x - 4
x + 3y = 12
(1, -2/3)
(-1, 2)
(6, 2)

19:
3x - y = 7
2x + 3y = 1
(2, -1)
(2, -3)
an infinite number of solutions
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
16.
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

3%2Ax-2%2Ay=12
7%2Ax%2B2%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-2%2Ay%29=%2812%29%2A7 Multiply the top equation (both sides) by 7
-3%2A%287%2Ax%2B2%2Ay%29=%288%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
21%2Ax-14%2Ay=84
-21%2Ax-6%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%29-14%2Ay-6%2Ay%29=84-24

%2821-21%29%2Ax-14-6%29y=84-24

cross%2821%2B-21%29%2Ax%2B%28-14-6%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%2F-20 Divide both sides by -20 to solve for y



y=-3 Reduce


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

3%2Ax-2%28-3%29=12 Plug in y=-3


3%2Ax%2B6=12 Multiply



3%2Ax=12-6 Subtract 6 from both sides

3%2Ax=6 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%286%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=2 Multiply the terms on the right side


So our answer is

x=2, y=-3

which also looks like

(2, -3)

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

3%2Ax-2%2Ay=12
7%2Ax%2B2%2Ay=8

we get



graph of 3%2Ax-2%2Ay=12 (red) 7%2Ax%2B2%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,-3). This verifies our answer.



17.
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=15
6%2Ax%2B10%2Ay=-30

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 6 to some equal number, we could try to get them to the LCM.

Since the LCM of 3 and 6 is 6, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -1 like this:

2%2A%283%2Ax%2B5%2Ay%29=%2815%29%2A2 Multiply the top equation (both sides) by 2
-1%2A%286%2Ax%2B10%2Ay%29=%28-30%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
6%2Ax%2B10%2Ay=30
-6%2Ax-10%2Ay=30

Notice how 6 and -6 and 30 and -10 add to zero (ie 6%2B-6=0 10%2B-10=0)

However 30 and 30 add to 60 (ie 30%2B30=60);


So we're left with

0=60


which means no value of x or y value will satisfy the system of equations. So there are no solutions


So this system is inconsistent



18.
Rearrange the first equation
-x+%2B+y+=+-+4
+x+%2B+3y+=+12
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

-1%2Ax%2B1%2Ay=-4
1%2Ax%2B3%2Ay=12

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 -1 and 1 to some equal number, we could try to get them to the LCM.

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

1%2A%28-1%2Ax%2B1%2Ay%29=%28-4%29%2A1 Multiply the top equation (both sides) by 1
1%2A%281%2Ax%2B3%2Ay%29=%2812%29%2A1 Multiply the bottom equation (both sides) by 1


So after multiplying we get this:
-1%2Ax%2B1%2Ay=-4
1%2Ax%2B3%2Ay=12

Notice how -1 and 1 add to zero (ie -1%2B1=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%28-1%2Ax%2B1%2Ax%29%2B%281%2Ay%2B3%2Ay%29=-4%2B12

%28-1%2B1%29%2Ax%2B%281%2B3%29y=-4%2B12

cross%28-1%2B1%29%2Ax%2B%281%2B3%29%2Ay=-4%2B12 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:

4%2Ay=8

y=8%2F4 Divide both sides by 4 to solve for y



y=2 Reduce


Now plug this answer into the top equation -1%2Ax%2B1%2Ay=-4 to solve for x

-1%2Ax%2B1%282%29=-4 Plug in y=2


-1%2Ax%2B2=-4 Multiply



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

-1%2Ax=-6 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ax=%28-6%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.


x=6 Multiply the terms on the right side


So our answer is

x=6, y=2

which also looks like

(6, 2)

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

-1%2Ax%2B1%2Ay=-4
1%2Ax%2B3%2Ay=12

we get



graph of -1%2Ax%2B1%2Ay=-4 (red) 1%2Ax%2B3%2Ay=12 (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 (6,2). This verifies our answer.



19.
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

3%2Ax-1%2Ay=7
2%2Ax%2B3%2Ay=1

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 2 to some equal number, we could try to get them to the LCM.

Since the LCM of 3 and 2 is 6, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -3 like this:

2%2A%283%2Ax-1%2Ay%29=%287%29%2A2 Multiply the top equation (both sides) by 2
-3%2A%282%2Ax%2B3%2Ay%29=%281%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
6%2Ax-2%2Ay=14
-6%2Ax-9%2Ay=-3

Notice how 6 and -6 add to zero (ie 6%2B-6=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%286%2Ax-6%2Ax%29-2%2Ay-9%2Ay%29=14-3

%286-6%29%2Ax-2-9%29y=14-3

cross%286%2B-6%29%2Ax%2B%28-2-9%29%2Ay=14-3 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:

-11%2Ay=11

y=11%2F-11 Divide both sides by -11 to solve for y



y=-1 Reduce


Now plug this answer into the top equation 3%2Ax-1%2Ay=7 to solve for x

3%2Ax-1%28-1%29=7 Plug in y=-1


3%2Ax%2B1=7 Multiply



3%2Ax=7-1 Subtract 1 from both sides

3%2Ax=6 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%286%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 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)

3%2Ax-1%2Ay=7
2%2Ax%2B3%2Ay=1

we get



graph of 3%2Ax-1%2Ay=7 (red) 2%2Ax%2B3%2Ay=1 (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.