Eighth Grade Math: Literal Equations, Variable Isolation, and Properties of Equality

Notebook Setup and Organization Guidelines

Record the date at the top of the paper: $08/2426$ .
Maintain a continuous thread of daily math notes on notebook paper throughout the week.
Work sequentially down the page each day (for instance, starting at the top on Wednesday and writing downwards).

Mathematical Vocabulary and Core Concepts

Literal Equation:
- Definition: An actual equation containing two or more variables.
- Meaning of "Literal": Refers to something actual, real, or "for real" (e.g., in everyday dialogue where someone asks "literally?" and the response is "yeah, for real").
- Examples of multi-variable mathematical expressions that form literal equations when set equal to something:
- $2x + 3y$
- $5x - 4y$
- Structural Requirement: Every equation must end with an equal sign ( $=$ ).
Basic Equation:
- Definition: A mathematical statement formed when a math problem is followed by an equal sign ( $=$ ).
- Presence of Variables: A basic equation does not require variables and can consist purely of numeric calculations.
- Examples:
- $3 + 5 = 8$
- $4 \times 2 = 8$
- Concrete Word Problem Context:
- Scenario: Billy has $3$ apples and receives $5$ more apples.
- Equation: $3 + 5 = 8$ total apples.
Variables:
- Definition: Symbols used to represent unknown values or numbers.
- Meaning of "Vary": To change. In mathematics, it refers to a number or range of numbers representing a value that changes depending on the context.
- Variable Notation: Commonly represented using lowercase letters such as $x$ , $y$ , $s$ , or contextual letters like $a$ for apples.
- Word Problem Context (Steve's Apples):
- Scenario: Steve begins with $5$ apples and ends up with $8$ total apples.
- Representation: Let $a$ represent the unknown quantity of added apples.
- Equation: $5 + a = 8$
- Solution: Solving for $a$ yields $a = 3$ apples.
Equations vs. Expressions:
- Directions reading "write and solve the equation" versus "write the equation and solve for $x$ ":
- An expression lacks an equal sign and cannot be solved without setting it equal to something.
- Rewriting a numeric expression as an equation: Rewriting $4 \times 2$ with its product produces the equation $4 \times 2 = 8$
- When a variable $x$ is attached (e.g., $4 \times x = 8$ ), solving requires determining the exact numerical value of $x$ .

Variable Isolation and Application to Word Problems

Principle of Variable Isolation:
- Solving an equation for a variable requires isolating that variable, meaning it must be placed completely by itself on one side of the equal sign.
Word Problem: Gym and Library Students:
- Given Data:
- Total number of students in third period: $15$
- Number of students who went to the gym in the morning: $7$
- Variable $s$ : Represents the unknown number of students who stayed in the library.
- Formulating the Equation:
- $15 - 7 = s$
- Analyzing Isolation and Solution:
- The variable $s$ is already isolated on the right side of the equal sign.
- Evaluating the arithmetic on the left side: $15 - 7 = 8$
- Result: $s = 8$ students stayed in the library.

Both-Sides Rule and Number Line Operations

Fundamental Rule of Solving Equations:
- Whatever operation is performed on the left side of an equation must also be performed on the right side to maintain equality when solving higher-level equations.
Number Line Construction:
- Construct a horizontal number line ranging from $0$ up to $15$ to the right, and extend it in the negative direction from $-1$ down to $-7$ .
Understanding Signs and Operations:
- The operation $15 - 7$ can be equivalently written as $15 + (-7)$ .
- To eliminate a negative value like $-7$ and bring it to $0$ , add $7$ (move $7$ units to the right on a number line).
Sample Problem: Solving $x - 7 = -6$ :
- Given Equation: $x - 7 = -6$
- Objective: Isolate the variable $x$
- Step 1: To cancel out $-7$ , add $7$ to both sides of the equation.
- $x - 7 + 7 = -6 + 7$
- Step 2: Simplify both sides.
- Left side: $-7 + 7 = 0$ , leaving $x$
- Right side: Starting at $-6$ on the number line and moving $7$ units to the right reaches $1$
- Final Solution: $x = 1$

Equations with Positive Integers and Solution Verification

Sample Problem: Solving $1 = w + 6$ :
- Given Equation: $1 = w + 6$
- Objective: Isolate the variable $w$
- Step 1: To cancel out the positive $6$ , subtract $6$ from both sides of the equation.
- $1 - 6 = w + 6 - 6$
- Step 2: Simplify both sides.
- Right side: $6 - 6 = 0$ , leaving $w$
- Left side: $1 - 6 = -5$
- Final Solution: $-5 = w$ (or $w = -5$ )
Methods for Verification:
- Method 1: Direct Substitution
- Substitute $-5$ back into the original equation for $w$ :
- $6 + (-5) = 1$
- Since $6 - 5 = 1$ , the solution is correct.
- Method 2: Number Line Traversal
- Start at $6$ on the number line and move $-5$ units (five spaces to the left): $6 \rightarrow 5 \rightarrow 4 \rightarrow 3 \rightarrow 2 \rightarrow 1$
- Landing on $1$ verifies that $w = -5$ .

Decimals and Formal Properties of Equality

Sample Problem: Solving $y + 3.4 = 0.5$ :
- Given Equation: $y + 3.4 = 0.5$
- Objective: Isolate the variable $y$
- Step 1: Subtract $3.4$ from both sides of the equation.
- $y + 3.4 - 3.4 = 0.5 - 3.4$
- Step 2: Calculate the resulting values.
- Left side: $3.4 - 3.4 = 0$ , leaving $y$
- Right side: $0.5 - 3.4 = -2.9$
- Final Solution: $y = -2.9$
Formal Definitions of Properties of Equality:
- Addition Property of Equality:
- Rule: Whenever a number is added to one side of an equation to help isolate a variable, the exact same number must be added to the opposite side to keep the equation balanced.
- Example Application: Solving $x - 7 = -6$ by adding $7$ to both sides.
- Subtraction Property of Equality:
- Rule: Whenever a number is subtracted from one side of an equation to help isolate a variable, the exact same number must be subtracted from the opposite side to keep the equation balanced.
- Example Application: Solving $1 = w + 6$ by subtracting $6$ from both sides, and solving $y + 3.4 = 0.5$ by subtracting $3.4$ from both sides.