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Chapter 11: Problem 103

In the following exercises, graph each equation. $$ x-y=3 $$

Short Answer

Expert verified
Graph the equation y = x - 3 by plotting (0, -3) and (1, -2), then drawing the line through these points.

Step by step solution

01

- Rearrange Equation

Rewrite the equation in slope-intercept form (y = mx + b). Start with the given equation: \[ x - y = 3 \] Add y to both sides and subtract 3 from both sides to get: \[ y = x - 3 \] This is now in the form y = mx + b.
02

- Identify the Slope and Y-Intercept

Identify the slope (m) and the y-intercept (b) from the equation y = x - 3. Here, the slope m is 1, and the y-intercept b is -3.
03

- Plot the Y-Intercept

Start on the y-axis and plot the y-intercept at (0, -3).
04

- Use the Slope to Find Another Point

From the y-intercept (0, -3), use the slope to find another point. Since the slope m is 1, rise 1 unit up and run 1 unit to the right. This gives the point (1, -2).
05

- Draw the Line

Draw a line through the points (0, -3) and (1, -2) to graph the equation.

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Key Concepts

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

slope-intercept form
When graphing linear equations, it is often easiest to start by writing the equation in slope-intercept form. This form of the equation of a line is given by: \( y = mx + b \)
Here, \( m \) represents the slope of the line, and \( b \) represents the y-intercept. This form is particularly useful because it directly reveals the slope and the y-intercept, making it easy to graph the equation.

In the given problem, the initial equation is: \( x - y = 3 \).
To convert this to slope-intercept form, we rearrange it as follows:
  • Start by adding \( y \) to both sides: \( x = y + 3 \)
  • Then, subtract 3 from both sides to isolate \( y \): \( y = x - 3 \)

Now the equation is in the form \( y = mx + b \), allowing us to easily identify the slope and y-intercept for graphing.
slope
The slope of a line ( \( m \) in the equation \( y = mx + b \)) represents how steep the line is.
It tells how much the y-coordinate changes for a given change in the x-coordinate.

The slope is sometimes described as 'rise over run,' where:
  • 'rise' is the change in y (vertical change)
  • 'run' is the change in x (horizontal change)

For the equation \( y = x - 3 \), the slope ( \( m \)) is 1. This means that for every 1 unit increase in x, y increases by 1 unit.

To graph the line using the slope, start at a known point on the line (such as the y-intercept), then use the slope to find additional points.

In this case, starting from the y-intercept (0, -3), move 1 unit up and 1 unit to the right to reach the point (1, -2).
y-intercept
The y-intercept of a line ( \( b \) in the equation \( y = mx + b \)) is the point where the line crosses the y-axis.
This is the value of y when x is 0.

In the equation \( y = x - 3 \), the y-intercept is -3.
This means that the point (0, -3) is on the line.

To graph the equation, start by plotting the y-intercept on the y-axis. This gives the first point on the graph.

From the y-intercept (0, -3), use the slope to find another point on the line. In this case, with a slope of 1, go up 1 unit and right 1 unit to plot another point (1, -2).

Finally, draw a straight line through these points to complete the graph of the equation.

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Most popular questions from this chapter

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