Equation Arena Review

Solutions follow on Pages 2, 3, & 4

Master Solution Key • Realm I

SYSTEMS OF EQUATIONS & INEQUALITIES (MODULE 7)

Quests 01 to 06

Quest 01: Classification Consistent & Indep.

System: \(3x + 5y = -5\) and \(6y + 15 = -15x\).
Step 1: Find slopes in slope-intercept form:
• Eq 1: \(5y = -3x - 5 \Rightarrow y = -\frac{3}{5}x - 1\) (\(m_1 = -0.6\)).
• Eq 2: \(6y = -15x - 15 \Rightarrow y = -\frac{5}{2}x - 2.5\) (\(m_2 = -2.5\)).
Conclusion: Unequal slopes (\(m_1 \ne m_2\)) mean the lines intersect at exactly one point.

10-Ft Glance Check: Look for \(m_1 = -3/5\) and \(m_2 = -5/2\) on the board.

Quest 02: Solution Count No Solution (0)

System: \(2y - 8 = -6x\) and \(3y + 9x = -12\).
Step 1: Solve each for \(y\):
• Line 1: \(2y = -6x + 8 \Rightarrow y = -3x + 4\).
• Line 2: \(3y = -9x - 12 \Rightarrow y = -3x - 4\).
Conclusion: Both lines share identical slope (\(m = -3\)) but have different y-intercepts (\(4 \ne -4\)). The lines are parallel.

10-Ft Glance Check: Look for \(y = -3x + 4\) and \(y = -3x - 4\) (parallel lines).

Quest 03: Shoes & Jacket Shoes: $22 • Jacket: $16

Equations: Let \(x\) = shoes, \(y\) = jacket.
\(x + y = 38\) and \(x = y + 6\).
Step 1: Substitute \((y + 6)\) for \(x\):
\((y + 6) + y = 38 \Rightarrow 2y + 6 = 38 \Rightarrow 2y = 32 \Rightarrow y = 16\).
Step 2: Solve for \(x\): \(x = 16 + 6 = 22\).
Check: \(\$22 + \$16 = \$38\) and \(\$22 - \$16 = \$6\).

10-Ft Glance Check: Look for boxed $22 (shoes) and $16 (jacket).

Quest 04: Substitution \((x, y) = (-5, -2)\)

System: \(3x - 2y = -11\) and \(-4x + y = 18\).
Step 1: Isolate \(y\) in Eq 2: \(y = 4x + 18\).
Step 2: Substitute into Eq 1:
\(3x - 2(4x + 18) = -11 \Rightarrow 3x - 8x - 36 = -11\)
\(-5x = 25 \Rightarrow x = -5\).
Step 3: \(y = 4(-5) + 18 = -20 + 18 = -2\).

10-Ft Glance Check: Look for boxed ordered pair (-5, -2).

Quest 05: Elimination \((x, y) = (10, -12)\)

System: \(9x + 11y = -42\) and \(3x + 4y = -18\).
Step 1: Multiply Eq 2 by \(-3\):
\(-9x - 12y = 54\).
Step 2: Add to Eq 1:
\((9x - 9x) + (11y - 12y) = -42 + 54 \Rightarrow -y = 12 \Rightarrow y = -12\).
Step 3: \(3x + 4(-12) = -18 \Rightarrow 3x = 30 \Rightarrow x = 10\).

10-Ft Glance Check: Look for boxed ordered pair (10, -12).

Quest 06: Inequalities Test (0,0) & (-5,6) are sol.

Conditions: \(y \ge -x\) AND \(2x + y \le 2\).
• \((0,0)\): \(0 \ge 0\) (True) & \(0 \le 2\) (True) \(\Rightarrow\) YES.
• \((-5,6)\): \(6 \ge 5\) (True) & \(2(-5)+6 = -4 \le 2\) (True) \(\Rightarrow\) YES.
• \((4,-5)\): \(-5 \ge -4\) is FALSE (\(-5 < -4\)) \(\Rightarrow\) NO.
Both inequalities must be satisfied.

10-Ft Glance Check: Look for (0,0)=YES, (-5,6)=YES, (4,-5)=NO on board.

Speed Check Tip: Give a quick dry-erase star on their board when verified so they can immediately erase and advance. Page 2 of 4

Master Solution Key • Realm II

EXPONENTS AND ROOTS (MODULE 8)

Quests 07 to 12

Quest 07: Power of a Monomial \(27x^6y^{15}\)

Expression: \((3x^2y^5)^3\).
Step 1: Apply power of a product rule: \((ab)^n = a^n b^n\).
\((3)^3 \cdot (x^2)^3 \cdot (y^5)^3\).
Step 2: Evaluate coefficient: \(3^3 = 27\).
Step 3: Multiply exponents: \(x^{2 \times 3} = x^6\), \(y^{5 \times 3} = y^{15}\).
Caution: Don't multiply base: \(3 \times 3 \ne 9\).

10-Ft Glance Check: Check that base is 27, not 9! Boxed: \(27x^6y^{15}\).

Quest 08: Negative Exponents \(\frac{1}{3a^7b^2}\)

Expression: \(\frac{5a^{-5}b^4}{15a^2b^6}\).
Step 1: Coefficients: \(\frac{5}{15} = \frac{1}{3}\).
Step 2: Subtract exponents:
\(a^{-5 - 2} = a^{-7} = \frac{1}{a^7}\).
\(b^{4 - 6} = b^{-2} = \frac{1}{b^2}\).
Result: Put terms with negative powers into denominator: \(\frac{1}{3a^7b^2}\).

10-Ft Glance Check: Ensure all variables are in denominator: \(\frac{1}{3a^7b^2}\).

Quest 09: Scientific Notation \(5.972 \times 10^{24}\text{ kg}\)

Value: 5,972 followed by 21 zeros (24 digits after 5).
Rule: Scientific notation format is \(a \times 10^n\) where \(1 \le a < 10\).
Step 1: Moving the decimal point 24 places left gives \(a = 5.972\).
Step 2: Power is positive: \(n = 24\).
Result: \(5.972 \times 10^{24}\) kg.

10-Ft Glance Check: Look for positive power of 24: \(5.972 \times 10^{24}\).

Quest 10: Rational Exponents \(8\)

Evaluate: \(\left(\frac{100}{25}\right)^{3/2}\).
Step 1: Divide inside parentheses:
\(\frac{100}{25} = 4\).
Step 2: Apply exponent \(4^{3/2}\):
The denominator 2 is the index (square root): \(\sqrt{4} = 2\).
The numerator 3 is the power: \(2^3 = 8\).

10-Ft Glance Check: Look for clean integer 8 boxed on board.

Quest 11: Radical Distributive \(60\sqrt{2} - 30\)

Expression: \(5\sqrt{3}(4\sqrt{6} - 2\sqrt{3})\).
Step 1: Distribute to first term:
\(5\sqrt{3} \cdot 4\sqrt{6} = 20\sqrt{18} = 20\sqrt{9 \cdot 2} = 20(3)\sqrt{2} = 60\sqrt{2}\).
Step 2: Distribute to second term:
\(5\sqrt{3} \cdot (-2\sqrt{3}) = -10\sqrt{9} = -10(3) = -30\).
Result: \(60\sqrt{2} - 30\).

10-Ft Glance Check: Look for \(60\sqrt{2} - 30\) (terms cannot be combined).

Quest 12: Common Base Equation \(x = 3\)

Equation: \(4^2 = 16^{x - 2}\).
Step 1: Rewrite 16 with base 4: \(16 = 4^2\).
\(4^2 = (4^2)^{x - 2} \Rightarrow 4^2 = 4^{2(x - 2)}\).
Step 2: Equate exponents:
\(2 = 2(x - 2) \Rightarrow 2 = 2x - 4 \Rightarrow 2x = 6 \Rightarrow x = 3\).
Check: \(16^{3-2} = 16^1 = 16 = 4^2\) ✓.

10-Ft Glance Check: Look for boxed x = 3 on board.

Zero exponent reminder: Make sure students know \((-5/6)^0 = 1\) if asked. Page 3 of 4

Master Solution Key • Realm III & Finale

EXPONENTIAL FUNCTIONS & BOSS BATTLE SOLUTIONS

Quests 13 to 18 + Bosses

Quest 13: Geometric Sequences A: No • B: Yes • C: Yes

A) \(2, 6, 10, 14 \dots\) difference \(d=+4\) \(\Rightarrow\) Arithmetic (No).
B) \(4, 16, 64, 256 \dots\) ratio \(r=4\) \(\Rightarrow\) Geometric (Yes).
C) \(50, 25, 12.5, 6.25 \dots\) ratio \(r=0.5\) \(\Rightarrow\) Geometric (Yes).

Quest 14: Explicit Nth Term \(a_n = \frac{1}{2}(16)^{n-1}\)

Sequence: \(\frac{1}{2}, 8, 128, 2048, \dots\)
First term \(a_1 = \frac{1}{2}\). Common ratio \(r = 8 \div (1/2) = 16\).
Formula: \(a_n = a_1(r)^{n-1} = \frac{1}{2}(16)^{n-1}\).

Quest 15: Decay Model \(y = 358,000(0.998)^t\)

Rate \(r = 0.2\% = 0.002\). Decay factor \(b = 1 - r = 0.998\).
Model: \(y = 358,000(0.998)^t\). Represents decay because base is between 0 and 1.

Quest 16: Compound Interest \$4,437.86

\(P = \$2500, r = 0.039, t = 15\).
\(A = 2500(1 + 0.039)^{15} = 2500(1.039)^{15} \approx \$4437.86\).

Quest 17: Transformations Left 3 • Reflect y-axis

1. \(j(x) = 2^{x+3}\): Shifted 3 units left.
2. \(h(x) = 2^{-x}\): Reflected across the y-axis.

Quest 18: Points Equation \(y = 4^x\)

Passes \((1, 4)\) and \((2, 16)\). Base \(b = \frac{16}{4} = 4\).
At \(x = 1\), \(a(4)^1 = 4 \Rightarrow a = 1\). Function: \(y = 4^x\).

Dual Whiteboard Boss Battle Solutions 250 pts each

Boss 1: Sandal & Shoe System: Equations: \(2.50x + 4y = 2450\) and \(5.00x + 4y = 3700\).
Subtracting equations yields \(2.50x = 1250 \Rightarrow x = 500\) sandals.
Substitute: \(2.50(500) + 4y = 2450 \Rightarrow 1250 + 4y = 2450 \Rightarrow 4y = 1200 \Rightarrow y = 300\) shoes.

Sandals = 500, Shoes = 300

Boss 2: Recursive to Explicit & 5th Term: Given \(a_1 = 45\) and \(a_n = 3a_{n-1}\) for \(n \ge 2\).
Part A: \(a_n = a_1(r)^{n-1} = 45(3)^{n-1}\).
Part B: \(a_5 = 45(3)^{5-1} = 45(3^4) = 45(81) = 3645\).

\(a_n = 45(3)^{n-1}\)  |  \(a_5 = 3,645\)

Whiteboard Remediation: Catch These Across the Room

Decay Rate vs Factor: If you see \(y = 358,000(0.2)^t\) on a board, tell the group: "Did the city drop to 20% or lose 0.2%?"

Marker Dominance: If one student is doing all the writing and thinking, gently enforce the rule: "Pass the marker to your teammate right now."

Distributing Exponents: If you see \(9x^6y^{15}\) instead of \(27x^6y^{15}\), point to the 3 and ask: "What is 3 to the 3rd power?"

Question 10 • Module 8 Rational Exponents

Evaluate without using a calculator:

\[\left(\frac{100}{25}\right)^{3/2}\]

Teacher Answer: 8  [\(\frac{100}{25} = 4\); \(4^{3/2} = (\sqrt{4})^3 = 2^3 = 8\)]

Question 11 • Module 8 Radical Operations

Multiply and simplify completely:

\(5\sqrt{3}(4\sqrt{6} - 2\sqrt{3})\)

Teacher Answer: \(60\sqrt{2} - 30\)  [\(20\sqrt{18} - 10(3) = 60\sqrt{2} - 30\)]

Question 12 • Module 8 Common Base Equations

Solve for \(x\) by finding a common base:

\(4^2 = 16^{x - 2}\)

Teacher Answer: \(x = 3\)  [\(4^2 = 4^{2(x - 2)} \Rightarrow 2 = 2x - 4 \Rightarrow 2x = 6\)]

Question 13 • Module 9 Geometric Sequences

Identify which sequences are Geometric (Yes / No):

A) \(2, 6, 10, 14, \dots\)

B) \(4, 16, 64, 256, \dots\)

C) \(50, 25, 12.5, 6.25, \dots\)

Teacher Answer: A: NO (diff \(+4\)) • B: YES (\(r = 4\)) • C: YES (\(r = 0.5\))

Question 14 • Module 9 Explicit Nth Term Formula

Write the explicit formula \(a_n = a_1(r)^{n-1}\) for:

\(\frac{1}{2}, 8, 128, 2048, \dots\)

Teacher Answer: \(a_n = \frac{1}{2}(16)^{n-1}\)  [\(a_1 = 1/2\), common ratio \(r = 16\)]

Question 15 • Module 9 Exponential Decay

A city's population is 358,000 and is decreasing at an annual rate of 0.2%.

Write the function \(y = a(b)^t\) and state the decay factor \(b\).

Teacher Answer: \(y = 358,000(0.998)^t\)  [\(b = 1 - 0.002 = 0.998\)]

Question 16 • Module 9 Compound Interest

Ellie's parents invest $2,500 at 3.9% interest compounded annually for 15 years.

Use \(A = P(1 + r)^t\). How much is in the account? Round to the nearest cent.

Teacher Answer: \$4,437.86  [\(2500(1.039)^{15} \approx 4437.86\)]

Question 17 • Module 9 Transformations

Parent function: \(f(x) = 2^x\). Describe each transformation:

1. \(j(x) = 2^{x + 3}\)

2. \(h(x) = 2^{-x}\)

Teacher Answer: 1) Shift 3 units LEFT • 2) Reflection across y-axis

Question 18 • Module 9 Function from Points

Write an exponential function in the form \(y = b^x\) passing through:

\((1, 4)\)   and   \((2, 16)\)

Teacher Answer: \(y = 4^x\)  [Base \(b = 16 \div 4 = 4\)]

Boss Battle 01 • Synthesis 250 Points

A company spends $2.50 to make sandals (\(x\)) and $4.00 to make running shoes (\(y\)). Normal month total: $2,450. In April, they double sandals and spend $3,700.

\(2.50x + 4y = 2450\)

\(5.00x + 4y = 3700\)

How many pairs of sandals and running shoes are made in a normal month?

Teacher Answer: Sandals = 500 pairs • Running Shoes = 300 pairs

Boss Battle 02 • Synthesis 250 Points

A sequence is defined recursively: \(a_1 = 45\) and \(a_n = 3a_{n-1}\) for \(n \ge 2\).

Part A: Write the explicit formula for \(a_n\).

Part B: Calculate the exact value of the 5th term (\(a_5\)).

Teacher Answer: Part A: \(a_n = 45(3)^{n-1}\) • Part B: \(a_5 = 3,645\)