Equation Solvers Solutions Guide
Algebra I • Practice Solutions & Strategy
Equation Solvers Solutions Guide
Independent & Guided Practice
Multi-Step Equations & Distribution
The 4-Phase Equation Solving Protocol Verify each step before moving forward
1. Simplify Sides Distribute & combine like terms
2. Collect Variables Move variables to one side
3. Isolate the Term Add or subtract constants
4. Solve & Check Multiply/divide & substitute
Problem 01 Distribution & Two-Step
\( 4(2x - 3) = 28 \)
1. Distribute \( 4 \): \( 8x - 12 = 28 \)
2. Add \( 12 \) to both sides: \( 8x = 40 \)
3. Divide by \( 8 \): \( x = 5 \)
Check: \( 4(2(5) - 3) = 4(7) = 28 \) ✓
Trap: Forgetting to multiply \( 4 \) by \( -3 \).
Problem 02 Variables on Both Sides
\( 7y - 5 = 3y + 19 \)
1. Subtract \( 3y \) from sides: \( 4y - 5 = 19 \)
2. Add \( 5 \) to both sides: \( 4y = 24 \)
3. Divide by \( 4 \): \( y = 6 \)
Check: \( 7(6) - 5 = 37 \); \( 3(6) + 19 = 37 \) ✓
Tip: Move the smaller term (\( 3y \)) to keep positive values.
Problem 03 Negative Signs & Combining
\( -2(3k + 1) + 5 = 15 \)
1. Distribute \( -2 \): \( -6k - 2 + 5 = 15 \)
2. Combine constants: \( -6k + 3 = 15 \)
3. Subtract \( 3 \) & divide by \( -6 \): \( k = -2 \)
Check: \( -2(3(-2) + 1) + 5 = -2(-5) + 5 = 15 \) ✓
Trap: \( 12 \div (-6) = -2 \). Watch sign rules!
Problem 04 Fraction Coefficient Strategy
\( \frac{2}{3}(m - 6) = 8 \)
1. Multiply by reciprocal \( \frac{3}{2} \): \( m - 6 = 8 \cdot \frac{3}{2} \)
2. Simplify product: \( m - 6 = 12 \)
3. Add \( 6 \) to both sides: \( m = 18 \)
Check: \( \frac{2}{3}(18 - 6) = \frac{2}{3}(12) = 8 \) ✓
Alt: Distribute first: \( \frac{2}{3}m - 4 = 8 \implies \frac{2}{3}m = 12 \).
Student Practice Reference • Solutions Sheet Page 1 of 2
Advanced Solutions & Error Analysis
Special cases, contextual modeling, and self-diagnostic audit
Solutions Guide • Page 2
Special Case Outcomes
Case A: Identity Equation All Real Numbers
\( 3(4x - 2) = 12x - 6 \)
Distribute left side:\( 12x - 6 = 12x - 6 \)
Subtract \( 12x \):\( -6 = -6 \) (Always True)
Conclusion: Every real number makes this true. Solution: Infinitely Many Solutions (\( \mathbb{R} \)).
Case B: Contradiction No Solution
\( 5(2w + 1) = 10w - 7 \)
Distribute left side:\( 10w + 5 = 10w - 7 \)
Subtract \( 10w \):\( 5 = -7 \) (False!)
Conclusion: Variables cancel yielding a false statement. Solution: No Solution (\( \emptyset \)).
Real-World Application Model Problem 05 • Break-Even Analysis
Scenario: Apex Logistics charges a flat \$15 service fee plus \$2.50 per mile. Metro Express charges a \$27 service fee plus \$1.25 per mile. At how many miles (\( m \)) will both delivery services charge the exact same total amount?
1. Algebraic Setup
\( 15 + 2.50m = 27 + 1.25m \)
Cost Apex = Cost Metro
2. Step-by-Step Solve
\( 1.25m = 12 \)
\( m = 12 \div 1.25 \)
\( m = 9.6 \text{ miles} \)
3. Contextual Verification
Apex: \( 15 + 2.5(9.6) = \$39.00 \)
Metro: \( 27 + 1.25(9.6) = \$39.00 \)
Cost equal at 9.6 mi
Error Detective: Spot the Misconception
Student Prompt: Solve \( 6x - (2x - 5) = 25 \)
Line 1: \( 6x - 2x - 5 = 25 \) ✘
Line 2: \( 4x - 5 = 25 \implies 4x = 30 \implies x = 7.5 \)
Diagnosis: The student failed to distribute the negative sign to \( -5 \). Subtracting a negative creates a positive: \( -(-5) = +5 \).
Correct Solve: \( 4x + 5 = 25 \implies 4x = 20 \implies x = 5 \).
Self-Check Mastery Matrix
Rate your execution before submitting your practice worksheet:
✓ Distributed sign and number to all inside terms
✓ Combined only terms with identical variable powers
✓ Substituted final value back into original equation
Target: 100% verification accuracy through substitution
Mastery Solutions & Error Diagnostic Reference Page 2 of 2
