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Chapter 2: Problem 29

Give the slope and y-intercept of each line, and graph it. $$y=3 x-1$$

Short Answer

Expert verified
The slope is 3, and the y-intercept is -1.

Step by step solution

01

- Identify the Slope-Intercept Form

Recognize that the equation is in the slope-intercept form, which is written as \(y = mx + b\). In this form, \(m\) represents the slope and \(b\) represents the y-intercept.
02

- Determine the Slope

Identify the slope (\(m\)) of the line from the equation. Here, \(m = 3\).
03

- Determine the Y-Intercept

Identify the y-intercept (\(b\)) of the line from the equation. Here, \(b = -1\).
04

- Plot the Y-Intercept

On a graph, plot the point (0, -1), which is the y-intercept.
05

- Use the Slope to Plot Another Point

Starting from the y-intercept (0, -1), use the slope \(3\) to determine another point. The slope means rise over run, so go up 3 units and right 1 unit to plot the point (1, 2).
06

- Draw the Line

Draw a straight line through the points (0, -1) and (1, 2) to represent the line \(y = 3x - 1\).

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Key Concepts

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

headline of the respective core concept
Graphing linear equations involves creating a visual representation of algebraic equations on a coordinate plane. This helps us understand the relationship between two variables. The equation given is in the slope-intercept form, which is written as \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) represents the y-intercept. Recognizing this form is the first step in graphing. By identifying the slope and the y-intercept, we can easily plot the equation on a graph.
headline of the respective core concept
The slope of a line tells us how steep the line is. It describes the rate of change between the variables. In the equation \(y = 3x -1\), the slope \(m\) is 3. This means that for every unit increase in \(x\), \(y\) increases by 3 units. Slope can be positive or negative.

  • A positive slope means the line rises from left to right.
  • A negative slope means the line falls from left to right.
  • Slope is calculated by \( m = \frac{rise}{run} \).
For our equation, we go up (rise) 3 units and over (run) 1 unit from any point on the line.
headline of the respective core concept
The y-intercept is where the line crosses the y-axis. It tells us the value of \(y\) when \(x\) is zero. In the equation \(y = 3x - 1\), the y-intercept \(b\) is -1. To plot the y-intercept:
  • Locate the y-axis (the vertical axis) on the graph.
  • Find the point where the line crosses the y-axis. In this case, it is (0, -1).
This point is a crucial step in graphing as it gives us a starting point to draw the line.
headline of the respective core concept
Plotting points involves marking specific coordinates on a graph. These points help us draw trends or specific lines. For our equation, after plotting the y-intercept (0, -1), we use the slope to find another point. From (0, -1):
  • Move up 3 units (rise).
  • Move 1 unit to the right (run).
This gives us the point (1, 2). After plotting these two points, you draw a straight line through them. This represents the graph of the equation \(y = 3x - 1\).

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Most popular questions from this chapter

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