Linear Mastery Review

Task 4.2 — Classify Number of Solutions:

Determine if the system has one solution, no solution, or infinite solutions. Justify:

\(\{3y = 6x + 15, \quad y = 2x - 1\}\)

Task 4.3 — Solve for the Intersection Coordinate:

Find \(y\)-value

Solve the linear system by substitution or elimination. State the exact \(y\)-coordinate where they intersect: \quad \(\begin{cases} 3x - y = 4 \\ 6x + 2y = 24 \end{cases}\)

Mission Wrap-Up: Self-Assessment Rate confidence: 1 (need help) to 4 (ready for exam)

S1: Context & Rates 1 • 2 • 3 • 4

S2: Equations & Errors 1 • 2 • 3 • 4

S3: Literal Formulas 1 • 2 • 3 • 4

S4: Systems of Eqns 1 • 2 • 3 • 4

Unit 2 Review Activity • Algebra 1 Page 2 of 2

Algebra 1 Review Solutions • Study Guide Page 1 of 2

Teacher & Student Solutions

Station Sprint Activity Solutions

Part 2: Worked Answers & Test Item Diagnostic Matrix

STATION SPRINT KEY

S1 Context Decoder Answers

1.1 Meaning of \(8p\): The total cost in dollars of the \(p\) pairs of gloves before tax.

1.2 Equation: Notebooks: \(\frac{\$12}{4} = \$3\); Pens: \(\frac{\$6}{3} = \$2\).
Equation: \(3x + 2y = 30\)

1.3 Verify \((30, 20)\): \(0.5(30) + 1.5(20) = 15 + 30 = 45\). Since \(45 = 45\), YES, it is a solution!

S2 Error Doctor Answers

2.1 Diagnose Mistake: First Error: Step 2.
The student added \(4x\) instead of subtracting \(4x\). Correct Step 2: \(5x + 12 = 47 \implies 5x = 35 \implies \mathbf{x = 7}\).

2.2 Solve for \(k\):

\(3(2k + 5) - 4(k + 2) = 17\)
\(6k + 15 - 4k - 8 = 17\)
\(2k + 7 = 17 \implies 2k = 10 \implies \mathbf{k = 5}\)

S3 Formula Vault Answers

3.1 Solve for \(w\): Method A: \(P = 6w + 48 \implies \mathbf{w = \frac{P - 48}{6}}\)
Method B: \(\frac{P}{6} = w + 8 \implies \mathbf{w = \frac{P}{6} - 8}\)

3.2 Graph Doctor Errors: 1. Placed intercept at \((4, 0)\) on \(x\)-axis instead of \((0, 4)\) on \(y\)-axis.
2. Slope is \(-3\) (down 3, right 1), but student went up 3, right 1 (\(+3\)).

S4 Systems Showdown Answers

4.1 System Setup: \(\mathbf{m = 4n}\) and \(\mathbf{\frac{1}{3}m + 3n = -21}\)

4.2 Solution Type: \(3y = 6x + 15 \implies y = 2x + 5\). Slopes are equal (\(m = 2\)) with different intercepts (\(5 \neq -1\)) \(\implies\) No Solution (Parallel).

4.3 Intersecting \(y\)-value: Multiply top by 2: \(6x - 2y = 8\). Subtract from \(6x + 2y = 24 \implies 4y = 16 \implies \mathbf{y = 4}\).

Test Alignment & Remediation Guide

Matches Attached Illuminate Exam

Algebra 1 Review Solutions • Station Sprint Activity Page 2 of 2

Algebra 1 Instructional Coaching • Unit 2 Test Review Page 1 of 1

A linear function \(g(x)\) passes through the points \((-2, 11)\) and \((4, -1)\). Construct the function equation in slope-intercept form:

F.IF.C.9

Comparing Different Representations

Algebraic vs Numerical / Verbal

Function A (Algebraic Formula):

\(f(x) = -3x + 14\)

Rate of Change: _______ • Initial Value: _______

Function B (Table Values):

Passes through \((0, 8)\) and \((2, 2)\).

Rate of Change: _______ • Initial Value: _______

Comparison Question: Which function has a greater rate of change? Which function has a greater output value at \(x = 3\)? Justify algebraically:

A.REI.D.11

Points of Intersection as Solutions

Connecting \(f(x) = g(x)\) to graphs

1. Explain the Concept:

Explain why the solution to the equation \(f(x) = g(x)\) is the \(x\)-coordinate of the point where the graphs of \(y = f(x)\) and \(y = g(x)\) intersect.

2. Find Intersection Point:

Let \(f(x) = 2x - 1\) and \(g(x) = -x + 8\). Solve \(f(x) = g(x)\) algebraically to find the intersection coordinate \((x, y)\):

Arizona Math Standards • Algebra 1 Course Review Page 2 of 2

Page 2: Detailed Solutions for F.LE.A.2, F.IF.C.9 & A.REI.D.11

STANDARDS KEY Part 2: F.LE • F.IF • A.REI

F.LE.A.2

Constructing Linear Functions — Solutions

Test Tie: Q7, Q8

From Table of Values:

Slope: \(m = \frac{11 - 5}{3 - 1} = \frac{6}{2} = \mathbf{3}\)

Intercept: \(5 = 3(1) + b \implies \mathbf{b = 2}\)

Rule: \(\mathbf{f(x) = 3x + 2}\)

Verify with \(x=5\): \(3(5)+2 = 17\) ✓

From Two Points \((-2, 11)\) & \((4, -1)\):

Slope: \(m = \frac{-1 - 11}{4 - (-2)} = \frac{-12}{6} = \mathbf{-2}\)

Intercept: \(-1 = -2(4) + b \implies -1 = -8 + b \implies \mathbf{b = 7}\)

Rule: \(\mathbf{g(x) = -2x + 7}\)

Verify with \((-2, 11)\): \(-2(-2)+7 = 11\) ✓

F.IF.C.9

Comparing Different Representations — Solutions

Cross-Representation Analysis

Function A: \(f(x) = -3x + 14\)

Rate of Change: \(\mathbf{-3}\) • Initial Value: \(\mathbf{14}\)

Function B: \((0, 8)\) and \((2, 2)\)

Rate of Change: \(\frac{2-8}{2-0} = \mathbf{-3}\) • Initial: \(\mathbf{8}\)

Comparison Conclusions & Algebraic Justification:

• Rate of Change: Both functions have the same rate of change (\(-3\)).
• Greater Output at \(x = 3\): \(f(3) = -3(3) + 14 = -9 + 14 = \mathbf{5}\). For Function B, \(g(x) = -3x + 8\), so \(g(3) = -3(3) + 8 = -9 + 8 = \mathbf{-1}\). Function A is greater because \(5 > -1\).

A.REI.D.11

Points of Intersection as Solutions — Solutions

Test Tie: Q12, Q13

1. Conceptual Explanation:

Full Credit Explanation: An intersection point \((x, y)\) lies on both graphs, meaning that input value \(x\) yields identical output values \(y\) in both functions. Therefore, solving \(f(x) = g(x)\) finds precisely the \(x\)-value where their outputs are equal.

2. Intersection of \(f(x)=2x-1\) & \(g(x)=-x+8\):

\(2x - 1 = -x + 8\)

\(3x - 1 = 8 \implies 3x = 9 \implies \mathbf{x = 3}\)

\(y = 2(3) - 1 = \mathbf{5}\)

Intersection Point: \(\mathbf{(3, 5)}\)

Recommended Grading Scale: 10 total items (10 pts each). 90-100% Mastery • 80-89% Proficient • <80% Targeted Intervention.

Aligned to AZ CCRS Math Standards

Arizona Standards Answer Key • Algebra 1 Page 2 of 2