Week 5 — Quiz (auto-graded) · Systems of Linear Equations & Inequalities

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective tested: Objective 4 — substitution, elimination, classifying systems, systems of linear inequalities.
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Module 5.

This is the human-readable quiz with its vetted answer key and feedback. The import-ready Classic QTI is in F-quiz-week-05-qti.xml. AI is not permitted on quizzes (course AI policy). Every numeric answer below is pre-computed and independently re-verified (Python w05_verify.py, PASS).

Blueprint

No trick questions; distractors target the Week 5 misconceptions named in the lecture outline.

Questions, key, and feedback

Q1 (MC). Solve by substitution: y = 2x and x + y = 9. What is the solution?
- A. (3, 6) ✅
- B. (2, 4)
- C. (4, 5)
- D. (1, 8)
Feedback: Substitute y = 2x into x + y = 9: x + 2x = 9 → 3x = 9 → x = 3; y = 2(3) = 6. Solution: (3, 6). Check: 6 = 2(3) ✓; 3 + 6 = 9 ✓. (B uses x = 2 by arithmetic error; C guesses without substituting; D forgets to substitute y.)

Q2 (MC). Solve by elimination: x + y = 10 and x − y = 4. What is the solution?
- A. (8, 2)
- B. (7, 3) ✅
- C. (6, 4)
- D. (5, 5)
Feedback: The y-coefficients are +1 and −1 — already opposites. Add: 2x = 14 → x = 7; 7 + y = 10 → y = 3. Solution: (7, 3). Check: 7 + 3 = 10 ✓; 7 − 3 = 4 ✓. (A uses x = 8, confusing the equations; C subtracts right sides incorrectly; D averages the constants.)

Q3 (MC). Solve by elimination: 2x + 3y = 12 and x − y = 1. What is the solution?
- A. (3, 2) ✅
- B. (2, 3)
- C. (4, 1)
- D. (1, 4)
Feedback: Multiply equation 2 by 3: 3x − 3y = 3. Add to equation 1: 5x = 15 → x = 3; 3 − y = 1 → y = 2. Solution: (3, 2). Check: 2(3) + 3(2) = 12 ✓; 3 − 2 = 1 ✓. (B swaps x and y; C is a sign error in back-substitution; D reverses both values.)

Q4 (MC). Classify the system: 2x + y = 5 and 4x + 2y = 10.
- A. Consistent independent (one solution)
- B. Inconsistent (no solution)
- C. Consistent dependent (infinitely many solutions) ✅
- D. Cannot be determined
Feedback: Equation 2 is exactly 2 × equation 1. Multiply equation 1 by −2 and add: 0 = 0 — a true statement — so the system is dependent (same line, infinitely many solutions). (A is wrong — there is no unique intersection; B confuses 0=0 with 0=k; D is never the answer when you can solve.)

Q5 (MC). Classify the system: x + y = 3 and x + y = 7.
- A. Consistent independent (one solution)
- B. Inconsistent (no solution) ✅
- C. Consistent dependent (infinitely many solutions)
- D. Cannot be determined
Feedback: Subtracting equation 1 from equation 2: 0 = 4 — a false statement — so the system is inconsistent (no solution; the lines are parallel, same slope −1, different y-intercepts 3 and 7). (A is wrong — the same left side with different right sides cannot intersect; C confuses 0=4 with 0=0.)

Q6 (MC). When two distinct lines intersect at exactly one point, the system of equations has how many solutions?
- A. Zero
- B. One ✅
- C. Two
- D. Infinitely many
Feedback: Two distinct lines that intersect share exactly one point — the intersection — so the system has one solution. (A = parallel lines; D = same line.)

Q7 (MC). Two numbers sum to 20 and differ by 4. What are the two numbers?
- A. 10 and 10
- B. 14 and 6
- C. 12 and 8 ✅
- D. 16 and 4
Feedback: Let a + b = 20 and a − b = 4. Adding: 2a = 24 → a = 12; b = 20 − 12 = 8. Check: 12 + 8 = 20 ✓; 12 − 8 = 4 ✓. 12 and 8. (A both equal — ignores the difference; B gives 14 − 6 = 8, not 4; D gives 16 − 4 = 12, not 4.)

Q8 (MC). Adult tickets cost $8 each and child tickets cost $5 each. A group bought 10 tickets total for $68. How many adult tickets were purchased?
- A. 4 adult, 6 child
- B. 6 adult, 4 child ✅
- C. 5 adult, 5 child
- D. 8 adult, 2 child
Feedback: Let a + c = 10 and 8a + 5c = 68. From the first: c = 10 − a. Sub: 8a + 5(10 − a) = 68 → 3a = 18 → a = 6; c = 4. Check: 6 + 4 = 10 ✓; 6(8) + 4(5) = 48 + 20 = 68 ✓. 6 adult, 4 child. (A has the counts reversed; C gives 5(8)+5(5)=65 ≠ 68; D gives too much revenue.)

Q9 (MC). Which of the following points is in the solution region of the system y > x AND y < 4?
- A. (1, 3) ✅
- B. (2, 2)
- C. (3, 5)
- D. (0, 0)
Feedback: Test each: (1, 3): 3 > 1 ✓ and 3 < 4 ✓ — satisfies both. (2, 2): 2 > 2? No — on the boundary, not strictly greater. (3, 5): 5 < 4? No — outside. (0, 0): 0 > 0? No — on the boundary. (B is on the boundary of y = x; C violates y < 4; D is on the boundary of y = x.)

Q10 (Matching). Match each system type to its geometric description.

Feedback: Consistent independent = lines meet at one point. Inconsistent = parallel lines, never meet. Dependent = same line, every point a solution. The algebra signal for inconsistent is 0 = k (false), not 0 = 0 (which signals dependent).

Answer key (quick reference)

Quality gate (self-checked, computer-verified): each single-answer item has exactly one correct option; the matching item pairs all four left entries one-to-one with the right entries. Arithmetic pre-computed and independently re-verified (w05_verify.py, PASS): Q1 y=2x,x+y=9→(3,6); Q2 x+y=10,x−y=4→(7,3); Q3 2x+3y=12,x−y=1→(3,2); Q4 2×eq1=eq2→dependent; Q5 0=4→inconsistent; Q6 one intersection→1 solution; Q7 a+b=20,a−b=4→(12,8); Q8 a+c=10,8a+5c=68→(6,4); Q9 (1,3): 3>1 and 3<4 both true. All checks PASS. QTI parse confirmation: F-quiz-week-05-qti.xml parses as imsqti_xmlv1p2 with 10 items.

Item-bank entries (for variants + the midterm/final)

All ten items are tagged course=MATH120 · week=5 · objective=4 · topic=systems-linear-equations-inequalities and deposited in Item Bank: Week 5 — Systems of Linear Equations & Inequalities. (Tags: q1 substitution, q2 elimination-no-multiply, q3 elimination-with-multiply, q4 classify-dependent, q5 classify-inconsistent, q6 classify-concept, q7 word-problem-system, q8 mixture-system, q9 system-inequalities-point, q10 classify-matching.)

Canvas placement block

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title           = "Week 5 Quiz — Systems of Linear Equations & Inequalities"
assignment_group = "Quizzes"
points_possible = 10
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due_offset_days = 6        # 6 days after module start (Sun Oct 4)
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provenance      = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"