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).
