Systems Showdown Test
Algebra 1 • Unit Assessment
Systems of Linear Equations
Substitution, Elimination & Real-World Applications
Total Score ______ / 30
Name:
Date:
Period:
Instructions: Show all algebraic steps clearly. State your solutions as ordered pairs \((x, y)\) where applicable. No credit will be given for unsupported answers.
A Method Fluency
8 Points Total (4 pts each)
1. Solve by Substitution 4 pts
\[\begin{cases} y = 3x - 5 \\ 2x + 3y = 29 \end{cases}\]
Solution:
2. Solve by Elimination 4 pts
\[\begin{cases} 4x + 3y = 22 \\ 2x - 5y = -2 \end{cases}\]
Solution:
B Strategic Reasoning
4 Points Total
3. System Analysis: \[\begin{cases} x - 2y = 8 \\ 3x + 5y = 2 \end{cases}\]
4 pts
Part A (1 pt): Which method (Substitution or Elimination) is more efficient to start with? Justify why using the coefficients of the equations.
Part B (3 pts): Solve the system using your chosen method. Show all work.
Solution:
Algebra 1 • Unit Assessment: Systems of Equations Page 1 of 2 • Continue to Page 2 →
Systems Showdown Test • Page 2: Applications & Modeling
Student Name:
C Real-World Modeling & Problem Solving
18 Points Total (6 pts each)
4. The Theater Ticket Conundrum 6 pts
The West High Drama Club sold a total of 250 tickets for their opening night performance, bringing in a total revenue of $1,850. Student tickets cost $5 each and adult tickets cost $9 each.
a) Define variables & set up system (2 pts):
Let s = ___________________
Let a = ___________________
System of Equations:
b) Solve showing all algebraic steps (3 pts):
c) Final Answer (1 pt):
5. Custom Apparel Cost Analysis 6 pts
Apex Apparel charges a $40 screen-setup fee plus $8 per printed shirt. Bold Prints charges a $15 setup fee plus $13 per printed shirt.
a) Write cost equations (2 pts):
Apex: C = ____________
Bold: C = ____________
c) Recommendation (1 pt):
If an art club needs 10 shirts, which shop is cheaper and by how much?
b) Solve for break-even point (3 pts):
Break-even Shirts: _____ Total Cost: $_____
6. Robotics Arena Dimensions 6 pts
The robotics team builds a rectangular competition field with a perimeter of 54 feet. The length of the field is 3 feet more than twice its width.
a) Write the system (2 pts):
Let \(L =\) length, \(W =\) width
c) Area Calculation (1 pt):
b) Solve using substitution (3 pts):
Length: _____ ft Width: _____ ft
Algebra 1 • Unit Assessment: Systems of Equations Page 2 of 2 • End of Test
Systems Showdown Answer Key
Teacher Resource Answer Key & Rubric
Systems of Linear Equations • Key
Comprehensive Step-by-Step Solutions • Total: 30 Points
Grading Scale A: 27–30 • B: 24–26 C: 21–23 • D: 18–20
A Method Fluency Solutions
8 Points Total
1. Substitution Solution 4 pts
\(y = 3x - 5\) & \(2x + 3y = 29\)
[+1] Substitute: \(2x + 3(3x - 5) = 29\)
[+1] Distribute & combine: \(2x + 9x - 15 = 29 \implies 11x = 44 \implies \mathbf{x = 4}\)
[+1] Back-substitute: \(y = 3(4) - 5 = 12 - 5 \implies \mathbf{y = 7}\)
Mistake alert: \(3 \cdot (-5) \neq +15\) [+1] (4, 7)
2. Elimination Solution 4 pts
\(4x + 3y = 22\) & \(2x - 5y = -2\)
[+1] Multiply Eq 2 by \(-2\): \(-4x + 10y = 4\)
[+1] Add equations: \(13y = 26 \implies \mathbf{y = 2}\)
[+1] Solve for \(x\): \(4x + 3(2) = 22 \implies 4x = 16 \implies \mathbf{x = 4}\)
Mistake alert: \(-2 \cdot (-2) = +4\) [+1] (4, 2)
B Strategic Reasoning Solution
4 Points Total
3. System Analysis Solution: \(x - 2y = 8\) & \(3x + 5y = 2\)
4 pts
Part A: Preferred Method Justification (1 pt) [+1 pt]
Exemplary Answer: "Substitution is more efficient because the first equation has a leading variable \(x\) with a coefficient of 1 (\(x = 2y + 8\)), which isolates cleanly without fractions."
(Full credit also accepted if student chose elimination and justified multiplying Eq 1 by \(-3\) to cancel \(x\)).
Part B: Step-by-Step Solution (3 pts) [+3 pts]
Substitution Steps:
1. Isolate: \(x = 2y + 8\) [+1]
2. Substitute into Eq 2: \(3(2y + 8) + 5y = 2\)
3. \(6y + 24 + 5y = 2 \implies 11y = -22 \implies \mathbf{y = -2}\) [+1]
Find Second Variable & Coordinate:
4. \(x = 2(-2) + 8 = -4 + 8 \implies \mathbf{x = 4}\)
5. Final Ordered Pair: (4, −2) [+1]
Check: \(3(4) + 5(-2) = 12 - 10 = 2\) ✓
Algebra 1 • Teacher Solutions & Scoring Rubric Page 1 of 2 • Page 2 Solutions →
Systems Showdown Answer Key • Page 2: Applications & Rubrics
Section C: 18 Points
4. Theater Ticket Conundrum (6 pts) 6 pts
a) Setup & Definitions (2 pts)
[+1] \(s =\) number of student tickets
\(a =\) number of adult tickets
[+1] System:
\(\begin{cases} s + a = 250 \\ 5s + 9a = 1850 \end{cases}\)
b) Solving & c) Conclusion (4 pts)
[+1] Multiply Eq 1 by \(-5\): \(-5s - 5a = -1250\)
