Lesson 9: Patterns, Tables, and Algebraic Rules

Property

In a relationship between two quantities, the independent variable is the quantity that is changed or controlled (the cause).
The dependent variable is the quantity that is measured or observed as a result (the effect).

Examples

Property

An equation in two variables shows the relationship between an independent variable (input) and a dependent variable (output).
These equations have the general form where one variable is expressed in terms of another, such as $y = mx + b$, $y = kx$, or other forms like $A = lw$. In these equations:

• One variable represents the input (independent variable)
• The other variable represents the output (dependent variable)
• Constants and coefficients determine how the variables relate to each other

Examples

Property

A relationship between two quantities, or variables, can be represented in different ways. A table shows specific pairs of values that follow a rule, while an equation describes that same rule algebraically.

Examples

Table: A table can show how the cost ($y$) relates to the number of tickets purchased ($x$).

Equation: The same relationship can be represented by the equation:

Table: A table can show how the total cost ($y$) relates to the number of notebooks purchased ($x$).

Equation:

1. Look at how the total cost changes as $x$ increases by 1:

→ Each notebook adds $2.

2. Notice that when $x = 1$, $y = 5$.
3. Combine these observations:

Explanation

A table and an equation can describe the same mathematical relationship. The table lists specific examples of the relationship, while the equation provides a general rule that works for any pair of values. In this lesson, you will learn how to find the pattern in a table to write its corresponding equation.