Formula Makeover Notes Algebra 1 Guided Notes
FORMULA MAKEOVER
Name: ______________________
Date: ______________________
TOPIC 1: UNLEASHING VARIABLES (LITERAL EQUATIONS)
Goal: Isolate a specific variable by undoing operations in reverse order. Think of it as peeling back layers of an onion.
The Variable Isolation Blueprint
To solve a literal equation for \(y\) , get \(y\) entirely by itself. Treat all other variables like constants! Use opposite operations to undo and rearrange terms.
01
Guided Problem 1: Step-by-Step Fill-Ins
Solve the literal equation for \(y\): \(3x - 4y = 6\)
Step A:
Isolate the \(y\) term. Subtract from both sides.
\(-4y = \) \(+ 6\)
Step B:
Divide every single term on both sides by the coefficient of \(y\), which is .
\(\frac{-4y}{\quad\quad} = \frac{-3x}{\quad\quad} + \frac{6}{\quad\quad}\)
Step C:
Simplify signs and fractions. A negative divided by a negative becomes positive!
\(y = \) \(x\)
02
Guided Problem 2: Two-Column Algebraic Proof
Solve the literal equation for \(y\): \(4 = 5x + 6y\)
Mathematical Steps / Statements
Justifications / Reasons
\(4 = 5x + 6y\)
Given Equation
\(4 - \) \(= 6y\)
Subtraction Prop. of Equality:
\(\frac{4 - 5x}{\quad\quad} = y\)
Division Prop. of Equality:
\(y = - \frac{5}{6}x + \)
Symmetric Property & Simplify:
Unit 2: Linear Transformation Suite Page 1 of 2
Algebra 1 Guided Notes
FORMULA MAKEOVER
Name: ______________________
Date: ______________________
TOPIC 2: STANDARDIZING LINES (STANDARD FORM)
Goal: Rearrange equations from Point-Slope or Slope-Intercept into Standard Form: \(Ax + By = C\) .
The Standard Form Code of Law
Rule 1: Both \(x\) and \(y\) must be on the left, equal to constant \(C\) on the right.Rule 2: No fractions or decimals! \(A\), \(B\), and \(C\) must be integers.Rule 3: The lead coefficient \(A\) must be positive (\(A \ge 0\)).
03
Guided Problem 3: Step-by-Step Fill-Ins
Write the equation in standard form: \(y - 2 = \frac{9}{2}(x + 4)\)
Step A:
Multiply the entire equation by the denominator to clear fractions: .
\(\cdot (y - 2) = 2 \cdot \frac{9}{2}(x + 4)\) → \(2y - \) \(= 9(x + 4)\)
Step B:
Distribute on the right side to clear parentheses.
\(2y - 4 = 9x + \)
Step C:
Move all variables to the left side and constants to the right. Simplify so \(A\) is positive!
\(9x - \) \(y = \)
04
Guided Problem 4: Two-Column Algebraic Proof
Write the equation in standard form: \(y + 16 = -\frac{21}{2}(x - 2)\)
Mathematical Steps / Statements
Justifications / Reasons
\(y + 16 = -\frac{21}{2}(x - 2)\)
Given Equation
\(2 \cdot (y + 16) = -21(x - 2)\)
Multiplication Property: Multiply by denominator .
\(2y + 32 = -21x + \)
Distributive Property: Distribute \(-21\) on the right.
\(21x + 2y + 32 = 42\)
Addition Property: Add to both sides.
\(+ 2y = \)
Subtraction Property: Subtract from both sides.
Unit 2: Linear Transformation Suite Page 2 of 2
Formula Makeover Practice Algebra 1 Independent Practice
FORMULA MAKEOVER PRACTICE
Name: ______________________
Date: ______________________
PART 1: VARIABLE ISOLATION BLUEPRINT
Instructions: Solve each literal equation for \(y\) . Show every step of your work clearly inside the blueprint boxes.
Problem 1 Solve for \(y\)
\(2x - 5y = 10\)
SHOW WORK HERE
\(y =\)
Problem 2 Solve for \(y\)
\(8 = 3x + 2y\)
SHOW WORK HERE
\(y =\)
Problem 3 Solve for \(y\)
\(-6x - 3y = 12\)
SHOW WORK HERE
\(y =\)
Problem 4 Solve for \(y\)
\(1 = 4x - 8y\)
SHOW WORK HERE
\(y =\)
Unit 2: Linear Transformation Suite Page 1 of 2
Algebra 1 Independent Practice
FORMULA MAKEOVER PRACTICE
Name: ______________________
Date: ______________________
PART 2: THE STANDARD CODE
Instructions: Convert each equation into Standard Form \(Ax + By = C\) where \(A, B, C\) are integers and \(A \ge 0\).
Problem 5 Standard Form
\(y - 4 = \frac{2}{3}(x + 6)\)
SHOW WORK HERE
\(Ax+By=C:\)
Problem 6 Standard Form
\(y + 5 = -\frac{7}{4}(x - 8)\)
SHOW WORK HERE
\(Ax+By=C:\)
Problem 7 Standard Form
\(y - 1 = -\frac{1}{2}(x + 3)\)
SHOW WORK HERE
\(Ax+By=C:\)
Problem 8 Standard Form
\(y + 12 = \frac{15}{2}(x - 4)\)
SHOW WORK HERE
\(Ax+By=C:\)
Unit 2: Linear Transformation Suite Page 2 of 2
Formula Makeover Exit Ticket Algebra 1 Quick Check
FORMULA MAKEOVER EXIT TICKET
Name: ______________________
Date: ______________________
PART 1: VARIABLE ISOLATION BLUEPRINT
Instructions: Solve each literal equation for \(y\) . Show all algebraic steps clearly inside the boxes.
Problem 1 Solve for \(y\)
\(5x - 2y = 15\)
SHOW WORK HERE
\(y =\)
Problem 2 Solve for \(y\)
\(-3 = 4x + 7y\)
SHOW WORK HERE
\(y =\)
Problem 3 Solve for \(y\)
\(3x - y = -9\)
SHOW WORK HERE
\(y =\)
Problem 4 Solve for \(y\)
\(10 = 2x - 6y\)
SHOW WORK HERE
\(y =\)
Unit 2: Linear Transformation Suite Page 1 of 2
Algebra 1 Quick Check
FORMULA MAKEOVER EXIT TICKET
Name: ______________________
Date: ______________________
PART 2: THE STANDARD CODE
Instructions: Convert each linear equation into Standard Form \(Ax + By = C\) where \(A, B, C\) are integers and \(A \ge 0\).
Problem 5 Standard Form
\(y - 3 = \frac{4}{5}(x + 10)\)
SHOW WORK HERE
\(Ax+By=C:\)
Problem 6 Standard Form
\(y + 8 = -\frac{3}{2}(x - 6)\)
SHOW WORK HERE
\(Ax+By=C:\)
Problem 7 Standard Form
\(y + 2 = \frac{1}{3}(x - 9)\)
SHOW WORK HERE
\(Ax+By=C:\)
Problem 8 Standard Form
\(y - 6 = -\frac{5}{4}(x + 4)\)
SHOW WORK HERE
\(Ax+By=C:\)
Unit 2: Linear Transformation Suite Page 2 of 2
Formula Makeover Answer Key Teacher Reference & Solutions
FORMULA MAKEOVER KEY
Practice & Exit Ticket
ANSWER KEY
PART 1: VARIABLE ISOLATION BLUEPRINT (ANSWERS)
Step-by-step worked solutions for isolating variable \(y\) in Practice Problems 1 to 4.
Problem 1 Solution Solved
\(2x - 5y = 10\)
Steps:
1. Subtract \(2x\) from both sides:
\(-5y = -2x + 10\)
2. Divide all terms by \(-5\):
\(y = \frac{-2x}{-5} + \frac{10}{-5}\)
\(y =\) \(\frac{2}{5}x - 2\)
Problem 2 Solution Solved
\(8 = 3x + 2y\)
Steps:
1. Subtract \(3x\) from both sides:
\(8 - 3x = 2y\)
2. Divide all terms by \(2\):
\(y = -\frac{3}{2}x + \frac{8}{2}\)
\(y =\) \(-\frac{3}{2}x + 4\)
Problem 3 Solution Solved
\(-6x - 3y = 12\)
Steps:
1. Add \(6x\) to both sides:
\(-3y = 6x + 12\)
2. Divide all terms by \(-3\):
\(y = \frac{6}{-3}x + \frac{12}{-3}\)
\(y =\) \(-2x - 4\)
Problem 4 Solution Solved
\(1 = 4x - 8y\)
Steps:
1. Subtract \(4x\) from both sides:
\(1 - 4x = -8y\)
2. Divide all terms by \(-8\):
\(y = \frac{-4}{-8}x + \frac{1}{-8}\)
\(y =\) \(\frac{1}{2}x - \frac{1}{8}\)
Unit 2: Linear Transformation Key Page 1 of 4
Teacher Reference & Solutions
FORMULA MAKEOVER KEY
Practice & Exit Ticket
ANSWER KEY
PART 2: THE STANDARD CODE (ANSWERS)
Step-by-step worked solutions for converting equations to Standard Form \(Ax+By=C\) in Practice Problems 5 to 8.
Problem 5 Solution Solved
\(y - 4 = \frac{2}{3}(x + 6)\)
Steps:
1. Multiply by 3: \(3(y-4) = 2(x+6)\)
\(3y - 12 = 2x + 12\)
2. Subtract \(2x\) and add \(12\):
\(-2x + 3y = 24\)
3. Multiply by \(-1\) (so \(A \ge 0\)):
\(SF:\) \(2x - 3y = -24\)
Problem 6 Solution Solved
\(y + 5 = -\frac{7}{4}(x - 8)\)
Steps:
1. Multiply by 4: \(4(y+5) = -7(x-8)\)
\(4y + 20 = -7x + 56\)
2. Add \(7x\) and subtract \(20\):
\(7x + 4y = 36\)
Note: \(A=7\) is positive. Fully simplified!
