The given consists of of . When using the to solve a system of equations, it is necessary to isolate a . In the second equation, it will be easiest to isolate
x.
⎩⎪⎪⎨⎪⎪⎧x+y+z=2x+2z=52x+y−z=-1(I)(II)(III)
⎩⎪⎪⎨⎪⎪⎧x+y+z=2x=5−2z2x+y−z=-1
With a variable isolated in one of the equations, we can substitute its into the remaining equations. In the final step of the simplification of these substitutions, our goal is to have yet another variable isolated.
⎩⎪⎪⎨⎪⎪⎧x+y+z=2x=5−2z2x+y−z=-1
⎩⎪⎪⎨⎪⎪⎧(5−2z)+y+z=2x=5−2z2(5−2z)+y−z=-1
⎩⎪⎪⎨⎪⎪⎧5−2z+y+z=2x=5−2z2(5−2z)+y−z=-1
⎩⎪⎪⎨⎪⎪⎧5−2z+y+z=2x=5−2z10−4z+y−z=-1
(I), (III): Add and subtract terms
⎩⎪⎪⎨⎪⎪⎧5−z+y=2x=5−2z10−5z+y=-1
⎩⎪⎪⎨⎪⎪⎧-z+y=-3x=5−2z10−5z+y=-1
⎩⎪⎪⎨⎪⎪⎧y=z−3x=5−2z10−5z+y=-1
This time, the
y-variable was isolated in the first equation. We can now substitute its equivalent expression into the third equation.
⎩⎪⎪⎨⎪⎪⎧y=z−3x=5−2z10−5z+y=-1
⎩⎪⎪⎨⎪⎪⎧y=z−3x=5−2z10−5z+(z−3)=-1
⎩⎪⎪⎨⎪⎪⎧y=z−3x=5−2z10−5z+z−3=-1
⎩⎪⎪⎨⎪⎪⎧y=z−3x=5−2z7−4z=-1
⎩⎪⎪⎨⎪⎪⎧y=z−3x=5−2z-4z=-8
⎩⎪⎪⎨⎪⎪⎧y=z−3x=5−2zz=2
The value of
z is
2. Substituting
2 for
z into the first and the second equations, we can find the values of
x and
y.
⎩⎪⎪⎨⎪⎪⎧y=z−3x=5−2zz=2
⎩⎪⎪⎨⎪⎪⎧y=2−3x=5−2(2)z=2
⎩⎪⎪⎨⎪⎪⎧y=2−3x=5−4z=2
(I), (II): Subtract terms
⎩⎪⎪⎨⎪⎪⎧y=-1x=1z=2
The solution to the system is the point
(1,-1,2). This is the singular point at which all three planes . Let's check our solution by substituting the values into the system.
⎩⎪⎪⎨⎪⎪⎧x+y+z=2x+2z=52x+y−z=-1(I)(II)(III)
(I), (II), (III): Substitute values
⎩⎪⎪⎪⎨⎪⎪⎪⎧1+(-1)+2=?21+2(2)=?52(1)+(-1)−2=?-1
(I), (II), (III): Multiply
⎩⎪⎪⎪⎨⎪⎪⎪⎧1+(-1)+2=?21+4=?52+(-1)−2=?-1
(I), (III): Remove parentheses
⎩⎪⎪⎪⎨⎪⎪⎪⎧1−1+2=?21+4=?52−1−2=?-1
(I), (II), (III): Add and subtract terms
⎩⎪⎪⎨⎪⎪⎧2=2 ✓5=5 ✓-1=-1 ✓
Since the substitution of our answers into the given equations resulted in three , we know that our solution is correct.