The given consists of of . Notice that the of
y in the first and the third equations are the of the coefficient of
y in the second equation; they will add to be
0. Let's use the to find a solution to this system.
⎩⎪⎪⎨⎪⎪⎧x−y+z=-1x+y+3z=-32x−y+2z=0(I)(II)(III)
We can start by adding the second equation to the first and the third equations to eliminate the
y-terms.
⎩⎪⎪⎨⎪⎪⎧x−y+z=-1x+y+3z=-32x−y+2z=0(I)(II)(III)
⎩⎪⎪⎨⎪⎪⎧x−y+z+(x+y+3z)=-1+(-3)x+y+3z=-32x−y+2z+(x+y+3z)=0+(-3)
(I), (III): Remove parentheses
⎩⎪⎪⎨⎪⎪⎧x−y+z+x+y+3z=-1−3x+y+3z=-32x−y+2z+x+y+3z=0−3
(I), (III): Add and subtract terms
⎩⎪⎪⎨⎪⎪⎧2x+4z=-4x+y+3z=-33x+5z=-3
Next, we will use our two equations that are only in terms of
x and
z to solve for the value of one of the . We will once again apply the Elimination Method, but this time it will be similar to when using it in a system with only two variables.
⎩⎪⎪⎨⎪⎪⎧2x+4z=-4x+y+3z=-33x+5z=-3
▼
(III): Solve by elimination
⎩⎪⎪⎨⎪⎪⎧-6x−12z=12x+y+3z=-33x+5z=-3
⎩⎪⎪⎨⎪⎪⎧-6x−12z=12x+y+3z=-36x+10z=-6
⎩⎪⎪⎨⎪⎪⎧-6x−12z=12x+y+3z=-36x+10z+(-6x−12z)=-6+(12)
⎩⎪⎪⎨⎪⎪⎧-6x−12z=12x+y+3z=-36x+10z−6x−12z=-6+12
⎩⎪⎪⎨⎪⎪⎧-6x−12z=12x+y+3z=-3-2z=6
⎩⎪⎪⎨⎪⎪⎧-6x−12z=12x+y+3z=-3z=-3
Now that we know that
z=-3, we can substitute it into the first equation to find the value of
x.
⎩⎪⎪⎨⎪⎪⎧-6x−12z=12x+y+3z=-3z=-3
⎩⎪⎪⎨⎪⎪⎧-6x−12(-3)=12x+y+3z=-3z=-3
⎩⎪⎪⎨⎪⎪⎧-6x+36=12x+y+3z=-3z=-3
⎩⎪⎪⎨⎪⎪⎧-6x=-24x+y+3z=-3z=-3
⎩⎪⎪⎨⎪⎪⎧x=4x+y+3z=-3z=-3
The value of
x is
4. Let's substitute both values into the second equation to find
y.
⎩⎪⎪⎨⎪⎪⎧x=4x+y+3z=-3z=-3
⎩⎪⎪⎨⎪⎪⎧x=44+y+3(-3)=-3z=-3
⎩⎪⎪⎨⎪⎪⎧x=44+y−9=-3z=-3
⎩⎪⎪⎨⎪⎪⎧x=4y−5=-3z=-3
⎩⎪⎪⎨⎪⎪⎧x=4y=2z=-3
The solution to the system is
(4,2,-3). This is the singular point at which all three planes .
Now we can check our solution by substituting the values into the system.
⎩⎪⎪⎨⎪⎪⎧x−y+z=-1x+y+3z=-32x−y+2z=0
(I), (II), (III): Substitute values
⎩⎪⎪⎪⎨⎪⎪⎪⎧4−2+(-3)=?-14+2+3(-3)=?-32(4)−2+2(-3)=?0
(I), (II), (III): Multiply
⎩⎪⎪⎪⎨⎪⎪⎪⎧4−2−3=?-14+2−9=?-38−2−6=?0
(I), (II), (III): Add and subtract terms
⎩⎪⎪⎨⎪⎪⎧-1=-1 ✓-3=-3 ✓0=0 ✓
Since the substitution of our answers into the given equations resulted in three , we know that our solution is correct.