Learn on PengienVision, Mathematics, Grade 8Chapter 2: Analyze and Solve Linear Equations

Lesson 7: Analyze Linear Equations: y = mx

In this Grade 8 enVision Mathematics lesson from Chapter 2, students learn how to analyze proportional relationships by identifying slope as the constant of proportionality and writing linear equations in the form y = mx. Students practice finding slope using rise over run from graphs and tables, writing equations from two points, and graphing lines through the origin with both positive and negative slopes.

Section 1

Slope Formula

Property

The slope of the line between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

m = \frac{y_2 - y_1}{x_2 - x_1}

This formula calculates the ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run).

Examples

Find the slope between $(2, 3)$ and $(7, 9)$ . Using the formula: $m = \frac{9 - 3}{7 - 2} = \frac{6}{5}$ .
Find the slope between $(-1, 5)$ and $(3, -3)$ . Using the formula: $m = \frac{-3 - 5}{3 - (-1)} = \frac{-8}{4} = -2$ .
Find the slope between $(-4, -2)$ and $(-6, 8)$ . Using the formula: $m = \frac{8 - (-2)}{-6 - (-4)} = \frac{10}{-2} = -5$ .

Explanation

The slope formula is a way to calculate rise over run without a graph. It finds the vertical distance between points $(y_2 - y_1)$ and divides it by the horizontal distance $(x_2 - x_1)$ to find the steepness.

Section 2

Interpreting Slope: Direction and Steepness

Property

Positive slopes correspond to lines that increase from left to right.
Negative slopes correspond to lines that decrease from left to right.
The larger the absolute value of the slope, the steeper the graph.

Examples

A line with slope $m = 2$ is steeper than a line with slope $m = \frac{1}{3}$ because $|2| > |\frac{1}{3}|$ .

A line with slope $m = -3$ is steeper than a line with slope $m = -1$ because $|-3| > |-1|$ . Both lines slant downwards.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Slope Formula

Property

The slope of the line between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

m = \frac{y_2 - y_1}{x_2 - x_1}

This formula calculates the ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run).

Examples

Find the slope between $(2, 3)$ and $(7, 9)$ . Using the formula: $m = \frac{9 - 3}{7 - 2} = \frac{6}{5}$ .
Find the slope between $(-1, 5)$ and $(3, -3)$ . Using the formula: $m = \frac{-3 - 5}{3 - (-1)} = \frac{-8}{4} = -2$ .
Find the slope between $(-4, -2)$ and $(-6, 8)$ . Using the formula: $m = \frac{8 - (-2)}{-6 - (-4)} = \frac{10}{-2} = -5$ .

Explanation

Section 2

Interpreting Slope: Direction and Steepness

Property

Examples

A line with slope $m = 2$ is steeper than a line with slope $m = \frac{1}{3}$ because $|2| > |\frac{1}{3}|$ .

A line with slope $m = -3$ is steeper than a line with slope $m = -1$ because $|-3| > |-1|$ . Both lines slant downwards.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Slope Formula

Property

The slope of the line between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

m = \frac{y_2 - y_1}{x_2 - x_1}

This formula calculates the ratio of the change in the y-coordinates (rise) to the change in the x-coordinates (run).

Examples

Find the slope between $(2, 3)$ and $(7, 9)$ . Using the formula: $m = \frac{9 - 3}{7 - 2} = \frac{6}{5}$ .
Find the slope between $(-1, 5)$ and $(3, -3)$ . Using the formula: $m = \frac{-3 - 5}{3 - (-1)} = \frac{-8}{4} = -2$ .
Find the slope between $(-4, -2)$ and $(-6, 8)$ . Using the formula: $m = \frac{8 - (-2)}{-6 - (-4)} = \frac{10}{-2} = -5$ .

Explanation

Section 2

Interpreting Slope: Direction and Steepness

Property

Examples

A line with slope $m = 2$ is steeper than a line with slope $m = \frac{1}{3}$ because $|2| > |\frac{1}{3}|$ .

A line with slope $m = -3$ is steeper than a line with slope $m = -1$ because $|-3| > |-1|$ . Both lines slant downwards.