/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 34 Use intercepts and a checkpoint ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 34

Use intercepts and a checkpoint to graph each equation. $$3 x-6 y=15$$

Short Answer

Expert verified
The x-intercept is (5,0), the y-intercept is (0,-2.5). After verifying, the checkpoint (1,1) is on the line. Hence, the graph of the equation \(3x - 6y = 15\) is a straight line passing through these points.

Step by step solution

01

Find the x-intercept

Set \(y=0\) and solve for \(x\) in the equation \(3x - 6y = 15\). This gives the x-intercept.
02

Find the y-intercept

Set \(x=0\) and solve for \(y\) in the equation \(3x - 6y = 15\). This gives the y-intercept.
03

Choose a checkpoint

Select a point not on the line obtained by using the intercepts. This point is referred to as the checkpoint. In this scenario, we can take (1,1) for ease.
04

Verify the checkpoint

Substitute the coordinates of the checkpoint back into the original equation. These should satisfy the equation, if not, recheck the calculations done previously.
05

Plot the intercepts and the checkpoint on the graph

Now plot these points on the graph (the x-intercept, the y-intercept, and the checkpoint). A straight line has now been defined by these points.
06

Draw the line

Draw a line through the three points to complete the graph of the equation.

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 the point at which the line crosses the x-axis. This is where the graph of the equation has a y-value of zero. To find the x-intercept algebraically, you set the y-variable to zero and solve for x.

For example, given the equation 3x - 6y = 15, to find the x-intercept, replace y with zero: 3x - 6(0) = 15, which simplifies to 3x = 15. Dividing both sides by 3, you find that x = 5. Therefore, the x-intercept is the point (5,0). This specific intercept can tell us that when a graph passes through this point, the value of y is null at the location where x equals 5.
Y-Intercept
Conversely, the y-intercept is found where a line crosses the y-axis; here the line has an x-value of zero. To locate the y-intercept on an equation, you set the x-variable to zero and solve for y.

Using the same equation, set x=0 in 3x - 6y = 15. The equation simplifies to -6y = 15. Dividing by -6 yields y = -2.5. The coordinates of the y-intercept are therefore (0,-2.5). A line will intersect the y-axis at this point, indicating that when x is zero, y will have a value of -2.5.
Checkpoint Method
The checkpoint method involves choosing a random point (not on the x or y-axes) and verifying that it satisfies the equation. This helps to confirm that you've correctly found the intercepts and can be very helpful in ensuring the accuracy of your graph.

For instance, using the checkpoint (1,1) for the equation 3x - 6y = 15 and substituting these values in gives us 3(1) - 6(1) = 3 - 6, which equals -3 and not 15, meaning the checkpoint does not lie on the line. This indicates either the checkpoint is incorrect, or the previous steps have errors. It is essential to select a suitable checkpoint and make sure all the points align perfectly on the graph to depict the line accurately.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

What is an \(x\) -intercept of a graph?

14 is \(25 \%\) of what number? (Section \(2.4,\) Example 6 )

Explain how to graph a linear equation of the form \(A x+B y=0\).

The line passes through \((2,4)\) and has the same \(y\) -intercept as the line whose equation is \(x-4 y=8\) The line passes through \((2,6)\) and has the same \(y\) -intercept as the line whose equation is \(x-3 y=18\)

Simplify: \(\quad 3\left(12 \div 2^{2}-3\right)^{2}\). (Section \(1.8,\) Example 6 )
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.