Self-Test 06 - Systems of Linear Equations

Check your understanding

This self-test lets you check your understanding of Session 06. It is purely formative: nothing is tracked or graded, and nobody sees your answers. Work through the Tasks first, then use this quiz to confirm the ideas have stuck. A wrong answer triggers a 10-second cooldown before you can retry, so think before you click.

Linear Equations & Solutions

1. Which of these equations is linear in its unknowns?

2. Is \((x, y) = (2, 3)\) a solution of \(\;x + y = 5\;\) and \(\;2x - y = 1\)?

3. Solve \(\;x + y = 10\;\) and \(\;x - y = 4\). Type the value of \(x\).

4. Solve \(\;x + 2y = 20\;\) and \(\;x - y = 5\). Type the value of \(y\).

Row Operations & Elimination

5. Which of these is NOT an allowed row operation on an augmented matrix?

6. Applying \(R_2 \to R_2 - 2R_1\) to \(\left[\begin{array}{cc|c} 1 & 2 & 5 \\ 2 & 1 & 4 \end{array}\right]\) turns the second row into:

7. During elimination a row collapses to \(\;0 = 5\). The system has:

8. During elimination a row collapses to \(\;0 = 0\). This signals:

9. A triangular system reads \(\;x + y + z = 9,\ \ y + z = 6,\ \ z = 4\). By back-substitution, type the value of \(x\).

How Many Solutions & Business

10. Two lines in a \(2 \times 2\) system are parallel but not identical. The system has:

11. Three planes of a \(3 \times 3\) system meet at a single point. The system has:

12. Demand is \(\;q = 100 - 2p\;\) and supply is \(\;q = 3p - 25\). Type the equilibrium price (a whole number).

Well done. You solved every question. You can tell a linear equation from a non-linear one, check a candidate solution, solve a system by substitution and elimination, apply the row operations, read off whether a system has one, none, or infinitely many solutions, and find a market equilibrium. You are ready for the Mock Exam.