/*! 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 23 Use intercepts and a checkpoint ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 23

Use intercepts and a checkpoint to graph each equation. $$6 x-9 y=18$$

Short Answer

Expert verified
The graph of the equation passing through the points X-intercept (3,0) and Y-intercept (0,-2). The point (1,1) is a checkpoint which does not satisfy the equation and therefore does not lie on the line.

Step by step solution

01

Find the x-intercept

Let y=0, we substitute y=0 into the equation and solve for x to get the x-intercept. So, we get \(6x - 9*0 = 18\), simplifies to \(6x = 18\), and upon dividing by 6 from both sides we get x=3. So, our x-intercept is (3,0).
02

Find the y-intercept

Similarly, let x=0, then substitute x=0 into the equation and solve for y to find the y-intercept. Thus, \(6*0 - 9y = 18\), which simplifies to \(-9y = 18\), and upon dividing by -9 from both sides, we get y=-2. So, our y-intercept is (0,-2).
03

Choose a checkpoint

Now, we choose a point not on the line made by the intercepts (i.e., neither (3,0) nor (0,-2)). The point (1,1) is a simple choice. Substituting x=1 and y=1 into the equation, we have \(6*1 - 9*1 = -3\), which is not equal to 18. Hence this point (1,1) does not lie on the line.
04

Graph the equation

Using the x and y intercepts (3,0) and (0,-2), we plot these points on the graph. Then draw a line passing through the intercept points to represent 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 graph is where the line crosses the x-axis. To find it, you set the y-coordinate to 0 and solve the equation for x. In the equation 6x - 9y = 18, when we set y = 0, it simplifies to 6x = 18. Dividing both sides by 6 gives us x = 3, so the x-intercept is (3,0). This is a pivotal point since it establishes where the line begins on the horizontal axis.

When practicing with x-intercepts, always remember:
  • Set y to zero before solving for x.
  • The resulting point will always have the form (x,0).
Understanding x-intercepts is fundamental in graphing linear equations as they give us one of the two necessary points to draw the line of the equation on a graph.
Y-Intercept
In contrast to the x-intercept, the y-intercept is where a line crosses the y-axis. Here, you'll set the x-coordinate to 0 and solve for y. Taking our earlier example, 6x - 9y = 18, and setting x = 0 we get -9y = 18. Dividing both sides by -9, the solution is y = -2, which gives us the y-intercept as (0,-2).

The y-intercept is equally important as the x-intercept because it marks the start of the line on the vertical axis. Keep in mind to:
  • Set x to zero before finding y.
  • The point obtained will have a (0,y) form.
Understanding both intercepts is crucial as these are the common points used to sketch the graph of a linear equation.
Algebraic Graphing
Moving forward, algebraic graphing is a technique used to visually represent solutions of equations on a coordinate plane. Once we have our x and y-intercepts from the equation 6x-9y=18, we plot these points on a Cartesian plane.

For algebraic graphing, following these steps ensures accurate representation:
  • Plot the x-intercept and y-intercept accurately on the graph.
  • Use a straightedge to draw a line through these intercepts, which is your linear equation.
  • If necessary, select an additional 'checkpoint' outside the intercepts to ensure the line is correct.
In our example, the checkpoint could verify if our line is correct, but since (1,1) doesn't satisfy the equation, it tells us this point isn't on our graphed line.
Linear Equations
Lastly, let's talk about linear equations. These equations form straight lines when graphed and typically look like ax + by = c, where a, b, and c are constants. They have a constant rate of change and no variable is raised to a power other than 1. By finding the x-intercept and y-intercept, you have all the necessary information to draw the line.

Key points to note about linear equations include:
  • They produce straight lines on a graph.
  • The slope-intercept form, y = mx + b, reveals the slope (m) and y-intercept (b) directly, which can also be useful in graphing the line quickly.
Our given equation of 6x - 9y = 18 is a classic linear equation and provides a straightforward example of how linear relationships are visually represented in graphing.

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

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((4,-7)\) and perpendicular to the line whose equation is \(x-2 y=3\)

Write an equation in slope-intercept form of the line satisfying the given conditions. The line passes through \((-6,4)\) and is perpendicular to the line that has an \(x\) -intercept of 2 and a \(y\) -intercept of \(-4\)

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 \(x-3 y=18\)

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((5,-9)\) and perpendicular to the line whose equation is \(x+7 y=12\)

In Exercises \(74-77\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Every line in the rectangular coordinate system has an equation that can be expressed in slope-intercept form.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.