Systems of linear equations (up to 3 unknowns) practice questions.

Solve the system of equations: x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2.

• (3, 2, 1)
• (1, 3, 2)
• (2, 1, 3)
• (1, 2, 3)

Answer: (1, 2, 3)

Adding the first and third equations gives 2x + 3y = 8; adding the second and third gives 3x + y = 5. Solving these gives x = 1, y = 2, and then z = 6 − 1 − 2 = 3.

Consider the system x + y + z = 2, x + 2y + 3z = 5, x + 3y + 5z = k. Determine the value of k for which the system is consistent.

• 5
• 8
• 3
• 10

Answer: 8

R2 − R1 gives y + 2z = 3 and R3 − R2 gives y + 2z = k − 5, so consistency requires k − 5 = 3, i.e. k = 8. The system then has infinitely many solutions.

A system of three linear equations in three unknowns has no solution, yet every pair of the three corresponding planes intersects in a line. Describe the configuration of the planes.

• A triangular prism: three parallel, distinct intersection lines
• All three planes are parallel to each other
• The three planes intersect in a single common line
• Two of the planes are parallel and the third one crosses both of them

Answer: A triangular prism: three parallel, distinct intersection lines

If two planes were parallel, that pair would not intersect at all; if the planes met in a common line there would be infinitely many solutions. The prism arrangement is the only one where each pair meets but no point lies on all three.

Row reduction of a system of three equations in x, y and z produces the final row 0x + 0y + 0z = 4. State what this shows about the system.

• z = 4
• The system has infinitely many solutions
• The system has no solution
• The unique solution is x = y = z = 4

Answer: The system has no solution

The row reads 0 = 4, which is impossible, so the equations are inconsistent and the system has no solution. Do not confuse it with the row 0 = 0, which indicates redundancy.

After row reduction, a system of three equations in three unknowns has two non-zero rows and one row reading 0 = 0. Determine the nature of the solution set.

• A single point
• A one-parameter family of solutions (a line)
• A two-parameter family of solutions (a plane)
• The empty set

Answer: A one-parameter family of solutions (a line)

Two independent equations in three unknowns leave 3 − 2 = 1 free parameter, so the solutions form a line. Geometrically the three planes share a common line.

For which value of a does the system x + 2y = 4, 3x + ay = 12 have infinitely many solutions?

• a = 2
• a = −6
• No such value of a exists
• a = 6

Answer: a = 6

With a = 6 the second equation is exactly 3 times the first (3x + 6y = 12), so the two lines coincide and every point on the line is a solution. For any a ≠ 6 the lines cross in exactly one point.