Identify the number of solutions of the system of linear equations.

Infinitely many solutions

(4y+9 ,  y  , -5y-7)

Step-by-step explanation:

  x+  y +z =  2

2x-3y +z = 11

-3x+2y-2z=-13

Ok so I'm going to eliminate a variable to find it later.

I'm going to multiply the top two equations by 2 because when I add each to the third the z variable will get eliminated.

So let's the multiplication by 2 part first:

2x +2y+2z=4

4x-  6y+2z=22

-3x+ 2y-2z=-13

Now adding equation 1 to 3 and then also adding equation 2 to 3:

-x+4y+0=-9

 x-4y+0 =9

-3x+2y-2z=-13

If I add equation 1 to 2 I get the following system:

0+0+0=0

-3x+2y-2z=13

So anything satisfying this last equation is a solution.

So we are looking for our infinitely many points in terms of y.

(x,y,z)

We need to write x and z in terms of y.

Let's solve for z:

-3x+2y-2z=-13

Add 3x on both sides:

     2y-2z=3x-13

Subtract 2y on both sides:

         -2z=3x-2y-13

Divide both sides by -2:

            z=(3x-2y-13)/-2

Multiply top and bottom by -1:

            z=(-3x+2y+13)/2

            z=(2y-3x+13)/2

Let's use the first equation:

x+y+z=2

We are going to solve this for z:

Subtract (x+y) on both sides:

z=2-(x+y)

z=2-x-y

Plug into the other equation we solve for z:

2-x-y=(2y-3x+13)/2

Let's solve for x since again we are trying to write x and z in terms of y:

Multiply both sides by 2:

4-2x-2y=2y-3x+13

Subtract 2y on both sides:

4-2x-4y=-3x+13

Subtract 13 on both sides:

-9-2x-4y=-3x

Add 2x on both sides:

-9-4y=-x

Multiply both sides by -1:

9+4y=x

Use symmetric property of equality:

x=9+4y

Use commutative property of addition:

x=4y+9

So now let's go back to that first equation we solved for z.

z=(2y-3x+13)/2

We are going to replace x with (4y+9):

z=(2y-3(4y+9)+13)/2

Distribute:

z=(2y-12y-27+13)/2

Combine like terms:

z=(-10y-14)/2

z=-5y-7

So the solution in terms of y is:

(4y+9 ,  y  , -5y-7).

Let's check it:

It must satisfy each of the following:

x+  y +z =  2

2x-3y +z = 11

-3x+2y-2z=-13

Checking equation 1:

(4y+9)+y+(-5y-7)  =2

Combine like terms:

(4y+y+-5y)+(9+-7)=2

(0y)+(2)=2

0+2=2

2=2

The point satisfies the first equation.

Checking equation 2:

2(4y+9)-3y+(-5y-7)=11

Distribute:

8y+18-3y-5y-7=11

Combine like terms:

(8y-3y-5y)+(18-7)=11

(0y)+11=11

0+11=11

11=11

The point satisfies this equation 2.

Checking last equation:

-3(4y+9)+2y-2(-5y-7)=-13

Distribute:

-12y-27+2y+10y+14=-13

Combine like terms:

(-12y+2y+10y)+(-27+14)=-13

(0y)+(-13)=-13

0+-13=-13

-13=-13

So it checks out for all three equations.

Therefore we have verified the following point is a solution to all equations:

(4y+9 ,  y  , -5y-7).