Equation Alchemists Slides
Equation Alchemists
Mastering the Art of One-Step Equations
Grade 7 Mathematics
Our Quest Today
1
Understand how to maintain balance in an algebraic equation.
2
Master the use of inverse operations to isolate variables.
3
Solve one-step equations involving all four operations (+, -, ×, ÷).
The Balance Scale
\(x + 5\)
\(12\)
An equation is a mathematical statement that two expressions are equal. Think of it as a scale that must always stay level!
The Golden Rule
Whatever you do to ONE SIDE of the equation...
You MUST do to the OTHER SIDE!
Inverse Operations: The "Undo" Button
Addition (+) Subtraction (-)
Multiplication (×) Division (÷)
To solve for a variable, we must isolate it by using the operation that "undoes" what is currently being done to it.
Solving Addition
Use Subtraction!
Equation:
\(x + 8 = 15\)
Subtract 8 from both sides.
\(x + 8 - 8 = 15 - 8\)
\(x = 7\)
Alchemist's Trial: Addition
"Solve for y to reveal the hidden essence."
\(y + 12 = 30\)
Operation to Use:
Subtraction (-)
Solution:
\(y = 18\)
Solving Subtraction
Use Addition!
Equation:
\(m - 5 = 11\)
Add 5 to both sides.
\(m - 5 + 5 = 11 + 5\)
\(m = 16\)
Alchemist's Trial: Subtraction
"Heal the rift by adding back what was lost."
\(z - 24 = 6\)
Operation to Use:
Addition (+)
Solution:
\(z = 30\)
Solving Multiplication
Use Division!
Equation:
\(4a = 24\)
Note: 4a means 4 times a
Divide both sides by 4.
\(\frac{4a}{4} = \frac{24}{4}\)
\(a = 6\)
Alchemist's Trial: Multiplication
"Divide the potion into equal measures."
\(7k = 56\)
Operation to Use:
Division (÷)
Solution:
\(k = 8\)
Solving Division
Use Multiplication!
Equation:
\(\frac{w}{3} = 9\)
Multiply both sides by 3.
\(3 \cdot \left(\frac{w}{3}\right) = 9 \cdot 3\)
\(w = 27\)
Alchemist's Trial: Division
"Combine the ingredients to find the whole."
\(\frac{b}{5} = 12\)
Operation to Use:
Multiplication (×)
Solution:
\(b = 60\)
The Truth Potion: Checking Your Work
Never wonder if your answer is right! Simply substitute your value back into the original equation.
Original: \(x + 10 = 25\)
If \(x = 15\)...
\(15 + 10 = 25\)
\(25 = 25\) ✓ CORRECT!
A true alchemist always verifies their brew!
Master Alchemist Challenge
Challenge 1
\(x + 100 = 250\)
Challenge 2
\(y - 15 = 45\)
Challenge 3
\(8z = 96\)
Challenge 4
\(\frac{w}{4} = 25\)
Recipe for Success
1
Identify the operation currently being done to the variable.
2
Apply the inverse operation to BOTH sides.
3
Simplify to isolate the variable.
4
Check your work by substituting.
Exit Ticket
Solve the following equation on your way out and explain which inverse operation you used!
\(\frac{x}{6} = 7\)
Alchemy Mastered!
You are now ready to solve any one-step equation.
Equation Alchemists Worksheet
Alchemist's Equation Log
Practice the Art of Inverse Operations
Apprentice:
Date:
If Addition (+) Use Subtraction (-)
If Subtraction (-) Use Addition (+)
If Multiplication (×) Use Division (÷)
If Division (÷) Use Multiplication (×)
Phase 1: Balancing the Hearth
Problem 1 Addition
\(x + 14 = 32\)
Answer:
Problem 2 Subtraction
\(m - 9 = 21\)
Answer:
Phase 2: Distilling Elements
Problem 3 Multiplication
\(6y = 42\)
Answer:
Problem 4 Division
\(\frac{w}{5} = 15\)
Answer:
Phase 3: The Truth Potion (Substitution)
Solve the equation below, then show your check by substituting the answer back in.
1. Solve the Equation:
\(12 + z = 40\)
\(z = \)
2. Check Your Work:
The Alchemist's Riddle
An alchemist is brewing a potion. They started with some amount of essence (\(e\)), added 25 drops, and ended up with exactly 80 drops. Write and solve an equation to find out how many drops they started with.
Equation:
Solution:
Equation Alchemists Answer Key
Alchemist's Master Key
Teacher Reference - Solutions & Methods
Phase 1 Solutions
Problem 1
\(x + 14 = 32\)
\(x + 14 - 14 = 32 - 14\)
\(x = 18\)
Problem 2
\(m - 9 = 21\)
\(m - 9 + 9 = 21 + 9\)
\(m = 30\)
Phase 2 Solutions
Problem 3
\(6y = 42\)
\(\frac{6y}{6} = \frac{42}{6}\)
\(y = 7\)
Problem 4
\(\frac{w}{5} = 15\)
\(5 \cdot \left(\frac{w}{5}\right) = 15 \cdot 5\)
\(w = 75\)
Phase 3 Solution & Check
Solution Steps:
\(12 + z = 40\)
\(12 - 12 + z = 40 - 12\)
\(z = 28\)
The Truth Check:
Substitute \(z = 28\)
\(12 + 28 = 40\)
\(40 = 40\) ✓
Riddle Answer Key
Equation:
\(e + 25 = 80\)
Solution:
\(e = 55\) drops
