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How to Find if a Table is Linear?

Published in Linearity Analysis 3 mins read

You can determine if a table represents a linear relationship by checking for a consistent pattern of change in the y-values relative to a consistent pattern in the x-values.

Identifying Linearity in Tables

A linear relationship is characterized by a constant rate of change, which means for every consistent increase in x, there's a consistent increase or decrease in y. This is most easily observed by finding an addition or subtraction pattern. Here’s how you can approach it:

Steps to Determine Linearity:

  1. Examine the x-values: Check if the x-values increase or decrease by the same amount each time.
    • If they do not increase or decrease by the same amount, look for a consistent multiplying or dividing pattern.
    • If there isn't a consistent pattern in the x-values, determining linearity this way is more complicated.
  2. Examine the y-values: Check if the y-values change by a constant amount each time.
    • If the x-values increase by 1 each time, then linearity is present if the y-values also increase or decrease by the same number each time.
    • If the x-values change consistently and the y-values change consistently, the table is linear.
  3. Consistent Change: If the y-values increase or decrease by a constant value for a constant change in x-values, the table is linear.

Examples:

  • Linear Table Example:

    x y
    1 5
    2 3
    3 1
    4 -1

    In this table:

    • The x-values increase by 1 each time.
    • The y-values decrease by 2 each time.
    • Therefore, this table is linear because there is a consistent change of -2 in the y-values for every consistent change of +1 in the x-values.

  • Non-Linear Table Example:

    x y
    1 2
    3 3
    5 5
    7 8

    In this table:

    • The x-values increase by 2 each time.
    • The y-values do not increase or decrease by the same amount each time.
    • Therefore, this table is not linear because there isn't a consistent change in the y-values for every consistent change in the x-values.

Key Insight:

As the reference stated, “The easiest way to determine of a table is linear is to find an adding or subtracting pattern.” Look for this consistent pattern for both x and y values to determine linearity.

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