Solve three-variable systems and classify intersections.
Example 1
Solve x+y+z=25,y+z=14,z=5. Find x.
Each colour is one equation, drawn as a plane. Rotate and hide individual planes to inspect how they meet. No solution point is calculated or marked. Plane portions are clipped to the viewing cube.Show worked answer
Answer: 11
Subtract the second equation from the first.
x=25−14=11.
Example 2
Consider the complete system x+y+z=3,y+z=6,z=5. How many simultaneous solutions does it have?
Each colour is one equation, drawn as a plane. Rotate and hide individual planes to inspect how they meet. No solution point is calculated or marked. Plane portions are clipped to the viewing cube.Show worked answer
Answer: Exactly one
The third equation fixes z=5; then the second fixes y=1.
The first fixes x=−3. All three values are determined, so there is exactly one solution.
Example 3
Solve x+4y=24,y+5z=30,4x+16y+z=101. Enter z.
Show worked answer
Answer: 5
Subtract 4 times the first equation from the third: z=101−4×24=5.
Back-substitution then gives y=5,x=4.
Systems of linear equations practice | Year 12 Specialist | Ascent Arcade