Problem 20 Find the $x$ - and $y$ -inte... [FREE SOLUTION]

Chapter 0: Problem 20

Find the $x$ - and $y$ -intercepts if they exist and graph the corresponding line. $$y=\frac{1}{3} x-1$$

Short Answer

Expert verified

x-intercept: (3,0); y-intercept: (0,-1); graph the line through these points.

Step by step solution

Find the x-intercept

The x-intercept occurs when the line crosses the x-axis, which means that the y-value is zero. Start by setting y to 0 in the equation. \[ 0 = \frac{1}{3}x - 1 \] Solve for x by adding 1 to both sides. \[ 1 = \frac{1}{3}x \] Multiply both sides by 3 to solve for x. \[ x = 3 \] Thus, the x-intercept is at (3,0).

Find the y-intercept

The y-intercept occurs when the line crosses the y-axis, which means that the x-value is zero at this point. Substitute x with 0 in the equation. \[ y = \frac{1}{3} \cdot 0 - 1 \] Simplify to find the y-intercept. \[ y = -1 \] Thus, the y-intercept is at (0, -1).

Plot the intercept points

On a graph, plot the x-intercept (3, 0) and the y-intercept (0, -1). These points are where the line will cross the axes.

Draw the line

Using a ruler, draw a straight line through the intercept points (3,0) and (0, -1). This line represents the equation $ y = \frac{1}{3} x - 1 $.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

x-intercept

The x-intercept of a line is a key concept in algebra and helps in understanding how a line crosses the x-axis on a graph. It occurs where the line meets or touches the x-axis, and at this point, the y-coordinate is zero. So, to find the x-intercept, you set the y-value to zero in the equation of the line and solve for x.

Here鈥檚 a simpler representation:

Take the linear equation, for example, $y = \frac{1}{3}x - 1$.
Set $y = 0$ to find the x-intercept, resulting in the equation $0 = \frac{1}{3}x - 1$.
Solve for x by adding 1 to both sides which leads to $1 = \frac{1}{3}x$.
Finally, multiply both sides by 3 to find $x = 3$.

Thus, the x-intercept is at the coordinate (3,0). It's the point that tells you where the line crosses the x-axis. In the context of graphing, this is crucial because it serves as one of the anchor points to draw the line.

y-intercept

The y-intercept is another fundamental concept when examining lines on a graph. It's the point where the line crosses the y-axis, and its x-coordinate is always zero. To find the y-intercept, you substitute zero in place of x in the line's equation, then solve for y.

Here's how to find it with the equation $y = \frac{1}{3}x - 1$:

Substitute $x = 0$ into the equation, which transforms it into $y = \frac{1}{3} \times 0 - 1$.
Simplify the equation, and you get $y = -1$.

So, the y-intercept is at point (0, -1). This point signifies where the line cuts through the y-axis, and is important when you're graphing because it acts as another anchor point to help position the line accurately on a graph.

graphing lines

Graphing lines is an essential skill in mathematics that allows you to visually represent equations on a coordinate plane. By graphing, you can easily understand relationships between variables. To graph a line, you need at least two points, although more can help ensure accuracy. The intercept points often provide a simple and direct way of plotting these key locations.

The steps to graph a line from the equation $y = \frac{1}{3}x - 1$ are:

First, determine the x-intercept (3,0) and the y-intercept (0,-1), as these are critical points where the line crosses each axis.
Plot these intercepts on a graph. The x-intercept means the line hits the x-axis at three units along. The y-intercept shows the line touches at minus one unit down.
Once these points are positioned, use a ruler to draw a straight line through them. This line represents the equation $y = \frac{1}{3}x - 1$.

Graphing like this makes it easier to see how changes in the equation affect the line's slope and direction. Additionally, it helps in visually interpreting the solutions to linear equations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find the \(x\) - and \(y\) -intercepts if they exist and graph the corresponding line. $$y=\frac{1}{3} x-1$$

Short Answer

Step by step solution

Find the x-intercept

Find the y-intercept

Plot the intercept points

Draw the line

Key Concepts

x-intercept

y-intercept

graphing lines

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Probability and Statistics

Statistics

Discrete Mathematics

Geometry

Mechanics Maths

Study anywhere. Anytime. Across all devices.