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Chapter 3: Problem 22

Graph equation by hand. \(y=x-1\)

Short Answer

Expert verified
Plot points (0, -1) and (1, 0) and draw the line through them.

Step by step solution

01

- Identify the slope and y-intercept

The given equation of the line is in the form of the slope-intercept form, which is \( y = mx + b \). Here, \( m = 1 \) is the slope and \( b = -1 \) is the y-intercept.
02

- Plot the y-intercept

To begin graphing, start by plotting the y-intercept on the graph. The y-intercept is where the line crosses the y-axis. For the equation \( y = x - 1 \), the y-intercept is -1. So, place a point at (0, -1) on the graph.
03

- Use the slope to find another point

The slope \( m = 1 \) means that for every increase of 1 in x, y increases by 1. Starting from the y-intercept (0, -1), move 1 unit to the right (x=1) and 1 unit up (y=0), which gives you another point (1, 0). Plot this point on the graph.
04

- Draw the line

With the two points plotted, (0, -1) and (1, 0), draw a straight line through them. This line represents the equation \( y = x - 1 \).

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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 very useful for graphing. This form is written as\( y = mx + b \). Here鈥檚 what each part stands for:
  • \( y \): The y-coordinate of any point on the line
  • \( x \): The x-coordinate of any point on the line
  • \( m \): The slope of the line
  • \( b \): The y-intercept of the line (where the line crosses the y-axis)
The slope tells us how steep the line is. The y-intercept tells us the starting point on the y-axis. For the equation \( y = x - 1 \), the slope (\( m \)) is 1, and the y-intercept (\( b \)) is -1.
plotting points
To graph a linear equation, you need to plot some points first. Here鈥檚 how to do it step-by-step:
  • Find the y-intercept, which is -1 for \( y = x - 1 \). Plot this point on the y-axis at (0, -1).
  • Use the slope to find other points. The slope of 1 means you move 1 unit right and 1 unit up to get from one point to the next.
  • Starting from (0, -1), move 1 unit right to x=1 and 1 unit up to y=0. Plot the point (1, 0).
These points will help you draw the line accurately.
drawing lines
Once you have your points plotted, drawing the line is straightforward:
  • Use a ruler to connect the points you plotted. This ensures your line is straight.
  • Extend the line in both directions and add arrows at both ends to show it continues indefinitely.
  • Make sure the line passes directly through all plotted points.
For the equation \( y = x - 1 \), you should now have a straight line passing through the points (0, -1) and (1, 0). This line represents all solutions to the equation on the graph.

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