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Linear Equations and Graphs

Linear Equations and Graphs

Author: Kyle Wooldridge, M.Ed.

Tags: linear relationships, slope, y-intercept, graphing, rate of change, tables, equations, representations, modeling, 6.EE.C.9, 7.RP.A.2, 8.EE.B.5, 8.EE.B.6, A.CED.A.2, A.REI.D.10

Concept Overview

A linear relationship describes a constant rate of change between two quantities. It can be modeled using an equation of the form:

\( y = mx + b \)

where:

  • \( m \) is the slope — the rate of change or steepness
  • \( b \) is the y-intercept — the value of \( y \) when \( x = 0 \)

What Makes a Relationship Linear?

A relationship is called linear when one quantity changes at a constant rate with respect to another. This steady change forms a straight line when graphed.

Linear relationships appear whenever two variables are connected by a fixed rule — like earning money at a constant hourly rate or measuring temperature over time.

Interpreting Graphs and Tables

Linear relationships can be represented in multiple formats: tables, graphs, and equations. Each format provides a different way to understand the same relationship.

  • Tables: Show pairs of values. Look for a constant rate of change in \( y \) as \( x \) increases.
  • Graphs: Show straight lines. The slope tells how steep, and the intercept tells where the line crosses the y-axis.
  • Equations: Tell the rule behind the pattern. \( y = mx + b \) shows the rate (\( m \)) and starting value (\( b \)).

Tip: To check if a relationship is linear, calculate the change in \( y \) over the change in \( x \) for several points. If the ratio is always the same, it’s linear.

Graphing the Linear Equation: \( y = 2x + 1 \)

Table of Values

\( x \) \( y = 2x + 1 \)
-2 \( y = 2 \cdot \textcolor{#0077cc}{-2} + 1 = -4 + 1 = -3 \)
-1 \( y = 2 \cdot \textcolor{#0077cc}{-1} + 1 = -2 + 1 = -1 \)
0 \( y = 2 \cdot \textcolor{#0077cc}{0} + 1 = 0 + 1 = 1 \)
1 \( y = 2 \cdot \textcolor{#0077cc}{1} + 1 = 2 + 1 = 3 \)
2 \( y = 2 \cdot \textcolor{#0077cc}{2} + 1 = 4 + 1 = 5 \)

Graph

-4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4

Understanding Slope and Intercept

A linear equation models a relationship between two variables — often written in slope-intercept form:

\( y = mx + b \)

  • Slope (\( m \)): The slope represents the rate of change. It is calculated as:

    \( m = \frac{y_2 – y_1}{x_2 – x_1} \)

    where \( (x_1, y_1) \) and \( (x_2, y_2) \) are any two points on the line. This is called the rise-over-run formula.
    If the slope is:
    • Positive: the line rises from left to right
    • Negative: the line falls from left to right
    • Zero: the line is flat (horizontal)
    • Undefined: vertical line (no run; \( x_2 – x_1 = 0 \))
  • y-intercept (\( b \)): The y-intercept is the point where the line crosses the y-axis. It tells the value of \( y \) when \( x = 0 \).
    This is often the “starting point” in real-world contexts — like an initial cost, balance, or amount.
    Example: In \( y = 2x + 3 \), the y-intercept is 3.

Visual Aid

Forms of Linear Equations & Conversions

Linear equations can appear in different forms depending on the context or the information available. Each form emphasizes a different part of the relationship between \( x \) and \( y \).

Slope-Intercept Form

\( y = mx + b \)

where:

  • \( m \) is the slope (rate of change)
  • \( b \) is the y-intercept (value of \( y \) when \( x = 0 \))

Standard Form

\( Ax + By = C \)

where \( A \), \( B \), and \( C \) are typically integers, \( A \) is non-negative, and \( B \) is not zero. This is useful for modeling constraints or working with systems of equations.

Point-Slope Form

If a line passes through a point \( (x_1, y_1) \) with slope \( m \), we can write:

\( y – y_1 = m(x – x_1) \)

This is especially helpful when the slope and a single point are known.

Converting Between Forms

  • From point-slope to slope-intercept: Expand and simplify.
  • From slope-intercept to standard form: Rearrange so \( Ax + By = C \).
  • From a table or graph: Find slope using any two points, then plug into slope-intercept or point-slope form.

Example: From Table to Equation

Given the points \( (1, 4) \), \( (2, 6) \), and \( (3, 8) \):

  • Change in \( y \): \( 6 – 4 = 2 \), \( 8 – 6 = 2 \) → Constant rate = 2
  • Use point-slope: \( y – 4 = 2(x – 1) \)
  • Simplify: \( y = 2x + 2 \)

Example: From Graph to Standard Form

If a graph passes through \( (0, 3) \) and \( (2, 7) \):

  • Find slope: \( \frac{7 – 3}{2 – 0} = 2 \)
  • Slope-intercept form: \( y = 2x + 3 \)
  • Move terms: \( -2x + y = 3 \)
  • Standard form: \( 2x – y = -3 \)

Try It Yourself!

  • Graph the equation \( y = -3x + 2 \)
  • Write an equation for a line with slope 5 and y-intercept -1
  • Create a table for \( y = x – 4 \) using: \( x = 0, 2, 4 \)
  • A bike rental costs $10 plus $3 per hour. Write an equation and find the cost for 5 hours.
Reveal Answers
  1. Line with slope -3, y-intercept 2; graph passes through (0, 2), (1, -1), (2, -4)
  2. \( y = 5x – 1 \)
  3. Table values for \( y = x – 4 \):
    \( x \) \( y = x – 4 \)
    0 \( y = 0 – 4 = -4 \)
    2 \( y = 2 – 4 = -2 \)
    4 \( y = 4 – 4 = 0 \)
  4. Equation: \( y = 3x + 10 \); for 5 hours: \( y = 3(5) + 10 = 25 \)

Connections & Extensions

Linear models appear in science (speed, temperature change), business (profit, cost), and technology (data usage).