Introduction to Linear Relations

Overview

Linear relations describe relationships between two variables that can be represented as a straight line on a graph. These relationships are typically expressed using linear equations of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The slope represents the rate of change between the variables, and the y-intercept represents the value of \( y \) when \( x = 0 \).

Key Concepts

  • Slope: The slope \( m \) indicates the steepness of the line. It is calculated as the change in \( y \) divided by the change in \( x \), often written as \( m = \frac{\Delta y}{\Delta x} \).
  • Y-Intercept: The y-intercept \( b \) is the point where the line crosses the y-axis, i.e., where \( x = 0 \).
  • Equation of a Line: A linear equation can be written in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
  • Graphing a Linear Equation: To graph a linear equation, identify the slope and y-intercept, and plot two points. Draw a straight line through these points to represent the equation.

Example of a Linear Equation

Consider the linear equation \( y = 2x + 3 \). This equation has a slope \( m = 2 \) and a y-intercept \( b = 3 \). This means that for every unit increase in \( x \), the value of \( y \) increases by 2, and when \( x = 0 \), \( y = 3 \).

Practice Problems

  1. Given the linear equation \( y = 4x - 5 \), identify the slope and the y-intercept.
    Solution

    The slope is \( 4 \) and the y-intercept is \( -5 \). This means that for every increase of 1 in \( x \), \( y \) increases by 4, and when \( x = 0 \), \( y = -5 \).

  2. Graph the equation \( y = -3x + 2 \).
    Solution

    To graph \( y = -3x + 2 \), plot the y-intercept \( (0, 2) \) and use the slope \( -3 \) to plot another point. The slope means that for every 1 unit increase in \( x \), \( y \) decreases by 3 units. The second point would be \( (1, -1) \). Draw a straight line through these points.

  3. Find the slope of the line that passes through the points \( (2, 3) \) and \( (5, 9) \).
    Solution

    The slope \( m \) is calculated as \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2 \). The slope of the line is 2.

  4. Write the equation of a line with a slope of \( -1 \) that passes through the point \( (4, 7) \).
    Solution

    Using the point-slope form \( y - y_1 = m(x - x_1) \), where \( m = -1 \), \( x_1 = 4 \), and \( y_1 = 7 \), the equation is \( y - 7 = -1(x - 4) \). Simplifying, we get \( y = -x + 11 \).

  5. Find the equation of a line that passes through the points \( (1, 2) \) and \( (3, 6) \).
    Solution

    First, find the slope: \( m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2 \). Then, use the point-slope form with point \( (1, 2) \) and slope \( 2 \): \( y - 2 = 2(x - 1) \). Simplifying, we get \( y = 2x \).

  6. Graph the equation \( y = 3x + 1 \).
    Solution

    To graph \( y = 3x + 1 \), plot the y-intercept \( (0, 1) \). Use the slope \( 3 \), meaning that for every 1 unit increase in \( x \), \( y \) increases by 3 units. Another point would be \( (1, 4) \). Draw a straight line through these points.

  7. Identify the slope and y-intercept of the equation \( 2y - 4x = 6 \).
    Solution

    Rewrite the equation in slope-intercept form \( y = mx + b \). First, divide by 2: \( y - 2x = 3 \), then solve for \( y \): \( y = 2x + 3 \). The slope is \( 2 \) and the y-intercept is \( 3 \).

→Representing Linear Relations