Representing Linear Relations

Overview

Linear relations can be represented in several ways: through tables, graphs, and equations. Understanding these different representations allows you to analyze and visualize the relationship between two variables. This section explores how to represent linear relations in various forms.

Key Methods of Representation

  • Table of Values: A table displays pairs of \( x \) and \( y \) values that satisfy the linear relation. These pairs can be plotted on a graph to form a straight line.
  • Graphing: The graph of a linear relation is a straight line. The slope (\( m \)) and y-intercept (\( b \)) are key elements in graphing the line.
  • Equation: The equation of a linear relation is typically in the form \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.

Steps for Graphing a Linear Relation

  1. Identify the slope \( m \) and y-intercept \( b \) from the equation.
  2. Plot the y-intercept on the graph, which is the point where the line crosses the y-axis.
  3. Use the slope to find another point on the line by following the "rise over run" pattern (how much the line goes up or down, and how much it goes left or right).
  4. Draw a straight line through the two points to complete the graph of the linear relation.

Practice Problems

  1. Write the equation of a line with a slope of 2 and a y-intercept of -3. Then, graph the line.
    Solution

    The equation of the line is: \( y = 2x - 3 \). To graph the line, start at the point (0, -3) on the y-axis, and use the slope of 2 to rise 2 units up and move 1 unit to the right for each point on the line.

  2. If the slope of a line is -1 and the y-intercept is 4, what is the value of \( y \) when \( x = 5 \)?
    Solution

    The equation is: \( y = -x + 4 \). Substituting \( x = 5 \) into the equation: \( y = -(5) + 4 = -5 + 4 = -1 \). The value of \( y \) when \( x = 5 \) is -1.

  3. Given the equation \( y = 3x + 6 \), plot the graph of this line. Identify the slope and the y-intercept.
    Solution

    The slope is 3 and the y-intercept is 6. Start by plotting the point (0, 6) on the y-axis. Then, for each unit increase in \( x \), rise 3 units to the right to find additional points on the line.

  4. A line passes through the points (2, 5) and (4, 9). Write the equation of the line.
    Solution

    First, calculate the slope using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) = (2, 5) \) and \( (x_2, y_2) = (4, 9) \). The slope is \( m = \frac{9 - 5}{4 - 2} = \frac{4}{2} = 2 \). Now use the point-slope form \( y - y_1 = m(x - x_1) \), which becomes \( y - 5 = 2(x - 2) \), simplifying to \( y = 2x + 1 \). The equation of the line is \( y = 2x + 1 \).

  5. Fill in the missing values in the following table of values for the equation \( y = 2x + 1 \):
    x y
    0 1
    1 3
    2 5
    3 7
  6. Graph the equation \( y = -x + 3 \) and identify the slope and y-intercept.
    Solution

    The slope is -1, and the y-intercept is 3. Start at the point (0, 3) and use the slope of -1 to rise 1 unit down and 1 unit right for each point on the line.

→The Cartesian Plane and Plotting Points