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Chapter 11: Problem 58

In the following exercises, graph by plotting points. $$ y=\frac{3}{2} x-3 $$

Short Answer

Expert verified
Plot points: (0, -3), (2, 0), and (4, 3). Draw a line through them.

Step by step solution

01

- Understand the Equation

The given equation is a linear equation in the slope-intercept form: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. In this case, \( m = \frac{3}{2} \) and \( b = -3 \).
02

- Identify the y-Intercept

The y-intercept is the point where the line crosses the y-axis. Set \( x = 0 \) in the equation to find the y-intercept: \[ y = \frac{3}{2} (0) - 3 = -3 \] So, the y-intercept is \( (0, -3) \).
03

- Choose x-Values to Find Corresponding y-Values

Choose several x-values and substitute them into the equation to find the corresponding y-values. For example: If \( x = 2 \), \[ y = \frac{3}{2} (2) - 3 = 3 - 3 = 0 \] If \( x = 4 \), \[ y = \frac{3}{2} (4) - 3 = 6 - 3 = 3 \] Thus, we also have the points \( (2, 0) \) and \( (4, 3) \).
04

- Plot the Points on the Graph

Plot the points found on a coordinate plane: \( (0, -3) \), \( (2, 0) \), and \( (4, 3) \).
05

- Draw the Line

Draw a straight line through the points plotted. This line represents the graph of the equation \( y = \frac{3}{2} x - 3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most useful and commonly used forms. It allows you to quickly understand the dynamics of a line. The general structure of a linear equation in slope-intercept form is: \ \ \[ y = mx + b \] \ \ Here, \ \ \[ m \] \ \ is the slope and \ \ \[ b \] \ \ is the y-intercept. The slope (\[ m \]), tells you how steep the line is. A higher slope means a steeper line.
The y-intercept (\[ b \]), shows you where the line crosses the y-axis. Understanding this form makes graphing equations simpler and faster.
Y-Intercept
The y-intercept is a key point in graphing linear equations. It shows where the line crosses the y-axis. For the equation \ \ \[ y = \frac{3}{2}x - 3 \], \ \ the y-intercept is found by setting \[ x = 0 \] and solving for \[ y \] like this:
  • \ \ \[ y = \frac{3}{2}(0) - 3 = -3 \]
This shows that the line crosses the y-axis at (0, -3). It's an anchor point to start plotting your graph. No matter the slope, the line must pass through this y-intercept. Knowing this helps you make sense of the line's position on the graph.
Plotting Points
Plotting points is essential for graphing linear equations. Once you have the y-intercept, you select other \ \ \[ x \] \ \ values to find corresponding \ \ \[ y \] \ \ values. This helps establish additional points the line will pass through. For example, using the equation: \ \ \[ y = \frac{3}{2}x - 3 \], \ \ choose \ \ \[ x = 2 \] \ \
  • \ \ \[ y = \frac{3}{2}(2) - 3 = 3 - 3 = 0 \]
giving you the point (2, 0). Select \ \ \[ x = 4 \]
  • \ \ \[ y = \frac{3}{2}(4) - 3 = 6 - 3 = 3 \]
giving you the point (4, 3). Plot these points, along with the y-intercept, on a graph. Each dot creates a visual guide for the line you are about to draw. The more points you plot, the more accurate your line will be.
Linear Equations
Linear equations form the foundation of graphing in algebra. These equations create straight lines when graphed. \ \ The equation \ \ \[ y = \frac{3}{2}x - 3 \] \ \ is a classic example of a linear equation. Here:
  • \[ m = \frac{3}{2} \] represents the slope, showing how much \[ y \] changes for a unit change in \ \ \[ x \]
  • \[ b = -3 \] is the y-intercept, showing where the line crosses the y-axis at (0, -3).
\[ y \] and \[ x \] are variables. \ \ Changing \[ x \] \ \ directly affects \ \ \[ y \], and vice versa. Graphing these equations helps visualize relationships between the variables.

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Most popular questions from this chapter

In the following exercises, find the slope of each line. $$ y=-1 $$

In the following exercises, graph the vertical and horizontal lines. $$ x=3 $$

In the following exercises, identify the most convenient method to graph each line. $$ y=\frac{1}{2} x+2 $$

Explain how you can graph a line given a point and its slope.

In the following exercises, use the slope formula to find the slope of the line between each pair of points. $$ (-5,-2),(3,2) $$
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