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Chapter 4: Problem 192

Graph using the intercepts. \(3 x+y=3\)

Short Answer

Expert verified
X-intercept (1,0); Y-intercept (0,3); Draw a line through these points.

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. For the equation \(3x + y = 3\): \[ 3x + 0 = 3 \Rightarrow x = 1 \] So, the x-intercept is (1,0).
02

- Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. For the equation \(3x + y = 3\): \[ 3(0) + y = 3 \Rightarrow y = 3 \] So, the y-intercept is (0,3).
03

- Plot the intercepts

Plot the points (1,0) and (0,3) on the coordinate plane. These points represent the x-intercept and y-intercept, respectively.
04

- Draw the line

Draw a straight line passing through the points (1,0) and (0,3). This line represents the graph of the equation \(3x + y = 3\).

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Key Concepts

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

x-intercept
In a linear equation, the x-intercept is where the line crosses the x-axis. At this point, the y-coordinate is always zero. To find the x-intercept for the equation given in the exercise, set y to 0 and solve for x. For example, in the equation \(3x + y = 3\), you would do the following:

\[3x + 0 = 3 \ \rightarrow x = 1\]
So the x-intercept is at the coordinate (1,0). You can always find the x-intercept by setting y to 0 and solving for x in any linear equation.
y-intercept
The y-intercept is the point where the line crosses the y-axis. Here, the x-coordinate is always zero. To find the y-intercept, set x to 0 and solve for y. Let's consider the same equation from the exercise: \(3x + y = 3\):

\[3(0) + y = 3\ \rightarrow y=3\]

So, the y-intercept is at (0,3). To find it in any linear equation, simply set x to 0 and solve for y. This will give you the point where your line intersects the y-axis.
plotting points
Plotting points involves placing points on the coordinate plane based on their coordinates (x,y). For example, to plot the points (1,0) and (0,3):
  • Find the x-coordinate first, then move left or right on the x-axis.
  • Next, find the y-coordinate and move up or down on the y-axis.

Consider the exercise:
  • For the x-intercept (1,0): Start at the origin (0,0), move 1 unit to the right and stay on the x-axis because y is 0.

  • For the y-intercept (0,3): Start at the origin, stay at x=0 and move 3 units up to the y-axis.
These steps will help you accurately place points on the coordinate plane for any linear equation.
coordinate plane
The coordinate plane is a 2-dimensional plane used for graphing equations. It has two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants.

Each point on the coordinate plane is represented by an ordered pair (x,y):
  • The first number (x) shows the position along the x-axis.
  • The second number (y) shows the position along the y-axis.
In the exercise, you plotted the points (1,0) and (0,3). This involves:
  • Reading the x and y values from the intercepts.
  • Placing points on the plane based on their coordinates.
  • Drawing a line through these points to graph the equation.
This is how you can visualize the solutions and understand how the linear equation behaves on the coordinate plane.

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Most popular questions from this chapter

Graph the linear inequality \(y \geq-2\)

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Graph the linear inequality \(y<4\)
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