Equation Sleuths Slides Algebra Foundations
Mission: Solve for \(x\)
Level 1 & Level 2 Equation Investigation
Equation Sleuths:
Cracking the Code of the Unknown Variable
Learn how to balance the scales, reverse operations, and isolate any variable like a master mathematician.
1
The Balance Scale
2
1-Step Equations
3
2-Step Strategies
The Golden Rule of Algebra
Principle #1
The Balance Law
An equation is a balanced scale . Whatever you do to one side of the equals sign, you must do to the other side.
Left Side = Right Side
Our Ultimate Objective
To isolate the variable (\(x\)). We want \(x\) completely alone on one side of the equation:
\(x = \text{number}\)
Key Memory Trick: An equation is like a mirror—treat both sides with equal fairness!
Inverse Operations: Undoing the Action
Investigator Toolkit
To liberate \(x\), we must undo whatever operation has been done to it using its mathematical opposite:
Addition
Undone by
−
Subtraction
Subtract both sides
×
Multiplication
Undone by
÷
Division
Divide both sides
Quick question for the class: What is the inverse operation of adding 14? Subtracting 14!
1
Level 1: One-Step (+ and −)
Single Inverse Action
Example A: Addition Undo with −
\(x + 8 = 21\)
\(-\; 8 \quad -\; 8\)
\(x = 13\)
Check: \(13 + 8 = 21\) ✓
Example B: Subtraction Undo with +
\(x - 12 = 19\)
\(+\; 12 \quad +\; 12\)
\(x = 31\)
Check: \(31 - 12 = 19\) ✓
Always substitute your solution back into the original equation to verify!
1
Level 1: One-Step (× and ÷)
Factors & Quotients
Example C: Multiplication Undo with ÷
\(6x = 42\)
\(\div\; 6 \quad \div\; 6\)
\(x = 7\)
Check: \(6(7) = 42\) ✓
Example D: Division (Fraction) Undo with ×
\(\frac{x}{5} = 8\)
\(\times\; 5 \quad \times\; 5\)
\(x = 40\)
Check: \(\frac{40}{5} = 8\) ✓
Remember: A coefficient stuck to \(x\) like \(6x\) means multiplication !
Interactive Partner Duel: Level 1
60-Second Challenge
Partner A: Solve on Whiteboard
\(x - 17 = 35\)
Identify operation applied to \(x\)
Apply inverse operation to both sides
Partner B: Solve on Whiteboard
\(\frac{x}{6} = 9\)
Identify operation applied to \(x\)
Apply inverse operation to both sides
Solution Reveal: Partner A: \(x = 52\) | Partner B: \(x = 54\)
Level 2: The Two-Step Strategy
Order of Operations in Reverse
The "Socks & Shoes" Rule
When getting dressed, you put on your socks first , then your shoes second .
When undressing, what must come off first? The shoes must come off before the socks!
Reverse PEMDAS (SADMEP)
1
First Undo Addition / Subtraction
Get rid of constant terms not attached to \(x\)
2
Second Undo Multiplication / Division
Detach the coefficient directly on \(x\)
Unwrap \(x\) like an onion: peel off outer layers before touching the core!
2
Solving \(3x + 5 = 26\) Step by Step
Standard Two-Step
1
Undo Addition
Subtract 5 from both sides to peel off the constant:
\(3x + 5 = 26\)
\(-\; 5 \quad -\; 5\)
\(3x = 21\)
2
Undo Multiply
Divide both sides by the coefficient 3:
\(3x = 21\)
\(\div\; 3 \quad \div\; 3\)
\(x = 7\)
3
Verification
Substitute \(x = 7\) back into original equation:
\(3(\mathbf{7}) + 5 \stackrel{?}{=} 26\)
\(21 + 5 = 26\)
\(26 = 26\) ✓ Solved!
Both steps followed the Balance Rule: whatever was done to left was matched on right!
2
Solving \(\frac{x}{4} - 3 = 7\)
Fraction Two-Step
1
Undo Subtraction First
Add 3 to both sides:
\(\frac{x}{4} - 3 = 7\)
\(+\; 3 \quad +\; 3\)
\(\frac{x}{4} = 10\)
2
Undo Division Second
Multiply both sides by 4:
\(\frac{x}{4} = 10\)
\(\times\; 4 \quad \times\; 4\)
\(x = 40\)
Spot-Check: \(\frac{40}{4} - 3 = 10 - 3 = 7\) Accurate & Confirmed!
Case of the Crooked Clue: Error Analysis
Spot the Flaw
Suspect's Solution: Contains Error!
Equation: \(5x - 4 = 26\)
Step 1: \(5x = 22\)
Step 2: \(x = 4.4\)
"I subtracted 4 from 26 and got 22!"
Class Debrief Questions
What was the suspect's fatal error in Step 1?
What should have been done to undo \(- 4\)?
What is the actual, true solution for \(x\)?
Correct Solution: \(5x = 30 \implies x = 6\)
Always pay close attention to the sign in front of the number!
The Equation Sleuth Code of Honor
Key Takeaways
1
Equal Balance
Apply every operation to both sides of the equals sign without exception.
Left = Right
2
Reverse Order
In two-step equations, undo addition/subtraction before multiplication/division.
Undo +/− first
3
Always Check
Plug your value of \(x\) back into the original problem to verify proof.
Substitute & Verify
Next Activity: Complete the Equation Sleuths Case File Worksheet!
Ready, Investigators? Let's Solve!
Equation Sleuths Worksheet Equation Sleuths Case File
Official Field Investigation: Solving for the Unknown Variable \(x\)
Case Level: 1 & 2
Name:
Date:
Period:
Investigator's Golden Rule & Inverse Operations
Addition (+)
Undo with Subtraction (−)
Subtraction (−)
Undo with Addition (+)
Multiplication (×)
Undo with Division (÷)
Division (÷)
Undo with Multiplication (×)
1 Part 1: Level 1 Investigations (One-Step Equations)
Show your inverse step on both sides & write your final answer.
Case #1: \(x + 14 = 31\)
Check: \((\quad) + 14 = 31\) \(x = \underline{\hspace{2.5cm}}\)
Case #2: \(y - 19 = 24\)
Check: \((\quad) - 19 = 24\) \(y = \underline{\hspace{2.5cm}}\)
Case #3: \(7a = 56\)
Check: \(7(\quad) = 56\) \(a = \underline{\hspace{2.5cm}}\)
Case #4: \(\frac{m}{6} = 8\)
Check: \(\frac{(\quad)}{6} = 8\) \(m = \underline{\hspace{2.5cm}}\)
Case #5: \(k + 27 = 15\)
Check: \((\quad) + 27 = 15\) \(k = \underline{\hspace{2.5cm}}\)
Case #6: \(-5w = 45\)
Check: \(-5(\quad) = 45\) \(w = \underline{\hspace{2.5cm}}\)
Equation Sleuths • Student Field File Page 1 of 2 — Turn over for Level 2 →
Level 2: Two-Step Case Files
Protocol: Undo Addition/Subtraction First → Then Undo Multiplication/Division
SADMEP Protocol
2 Solve Each Two-Step Equation & Verify Your Solution
2 steps required per case
Case #7: \(3x + 8 = 29\)
Check: \(3(\quad) + 8 = 29\) \(x = \underline{\hspace{2.2cm}}\)
Case #8: \(5p - 7 = 38\)
Check: \(5(\quad) - 7 = 38\) \(p = \underline{\hspace{2.2cm}}\)
Case #9: \(\frac{n}{4} + 9 = 15\)
Check: \(\frac{(\quad)}{4} + 9 = 15\) \(n = \underline{\hspace{2.2cm}}\)
Case #10: \(-2x + 13 = 1\)
Check: \(-2(\quad) + 13 = 1\) \(x = \underline{\hspace{2.2cm}}\)
Part 3: Interrogation Room — Spot the Imposter Error
Find & Fix the Mistake
Suspect solved: \(4x - 6 = 26\)
Step 1: \(4x = 20\) (Subtracted 6 from 26)
Step 2: \(x = 5\) (Divided by 4)
What was the suspect's error?
Correct Solution Workspace:
Correct \(x = \underline{\hspace{2cm}}\)
Part 4: Real-World Case Study
A detective purchases a specialized fingerprint kit for
Equation Sleuths Solutions Guide Solutions & Facilitation Guide
Master Key: Equation Sleuths Level 1 & Level 2 Practice
Teacher Reference Only
Instructional Goal: Mastery of variable isolation through inverse operations and balance preservation. Grades 6–9 Algebra
Facilitator Watch-Outs for Level 1:
Watch for students operating only on one side of the equal sign, or performing the operation written rather than its inverse (e.g., adding 14 in Case #1 instead of subtracting 14). Emphasize drawing a vertical "balance line" down through the equal sign.
1 Part 1 Master Solutions: Level 1 One-Step Equations
Case #1: \(x + 14 = 31\) Subtract 14
\(x + 14 - 14 = 31 - 14\)
\(x = 17\)
Check: \(17 + 14 = 31\) ✓ Valid
Case #2: \(y - 19 = 24\) Add 19
\(y - 19 + 19 = 24 + 19\)
\(y = 43\)
Check: \(43 - 19 = 24\) ✓ Valid
Case #3: \(7a = 56\) Divide by 7
\(\frac{7a}{7} = \frac{56}{7}\)
\(a = 8\)
Check: \(7(8) = 56\) ✓ Valid
Case #4: \(\frac{m}{6} = 8\) Multiply by 6
\(\frac{m}{6} \cdot 6 = 8 \cdot 6\)
\(m = 48\)
Check: \(\frac{48}{6} = 8\) ✓ Valid
Case #5: \(k + 27 = 15\) Subtract 27
\(k + 27 - 27 = 15 - 27\)
\(k = -12\)
Check: \(-12 + 27 = 15\) ✓ Valid (Integer check)
Case #6: \(-5w = 45\) Divide by −5
\(\frac{-5w}{-5} = \frac{45}{-5}\)
\(w = -9\)
Check: \(-5(-9) = 45\) ✓ Valid
Equation Sleuths • Answer Key & Guide Page 1 of 2 →
Level 2 Solutions & Case Analysis Guide
Two-Step Master Key
2 Cases 7–10 Step-by-Step Solutions
Case #7: \(3x + 8 = 29\) \(x = 7\)
1. Subtract 8: \(3x = 21\)
2. Divide by 3: \(x = 7\)
Check: \(3(7) + 8 = 21 + 8 = 29\) ✓
Case #8: \(5p - 7 = 38\) \(p = 9\)
1. Add 7: \(5p = 45\)
2. Divide by 5: \(p = 9\)
Check: \(5(9) - 7 = 45 - 7 = 38\) ✓
Case #9: \(\frac{n}{4} + 9 = 15\) \(n = 24\)
1. Subtract 9: \(\frac{n}{4} = 6\)
2. Multiply by 4: \(n = 24\)
Check: \(\frac{24}{4} + 9 = 6 + 9 = 15\) ✓
Case #10: \(-2x + 13 = 1\) \(x = 6\)
