/*! 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 25 Use intercepts and a checkpoint ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 25

Use intercepts and a checkpoint to graph equation. \(-x+4 y=6\)

Short Answer

Expert verified
The line representing the equation \(-x + 4y = 6\) will pass through the three points (-6, 0), (0, 1.5), and (1, 1.75). So these are the intercepts and one checkpoint used to graph this equation.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for x. This will give the point at which the graph intersects the x-axis. So, when y is zero, the equation becomes: \(-x + 4 * 0 = 6\). Solving this equation for x gives \(x = -6\). So, the x-intercept is at (-6, 0)
02

Find the y-intercept

The y-intercept is found in a similar manner by setting \(x = 0\) in the equation. Hence, the equation becomes: \(0 + 4y = 6\). Solving this equation for y gives \(y = 1.5\). So, the y-intercept is at (0, 1.5).
03

Choose a check point

Now that we have the intercepts, we can choose a checkpoint. It鈥檚 a good choice to take a point that is convenient to use. The origin (0,0) is often a good checkpoint to start from if it is not on the line. However, in this case the y-intercept is at (0, 1.5), so the point (0,0) is on the line. A suitable point could be x = 1. Replace x = 1 in the original equation: \(-1 + 4y = 6\), solving this for y gives \(y=7/4\) or 1.75. So, the point (1, 1.75) is on the graph.
04

Graph the function

Plot the intercepts and the checkpoint on the coordinate plane, and draw a line that passes through them to form the graph. By connecting the points (-6, 0), (0, 1.5), and (1, 1.75), graph the function. The line drawn should pass through these three points and extend beyond them in both directions.

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.

Understanding the X-Intercept
The x-intercept is a fundamental concept when working with linear equations. It represents the point where the graph of the equation crosses the x-axis. To find this point, you set the value of y to zero and solve the equation for x.

In our example equation, \[ -x + 4y = 6 \] we find the x-intercept by substituting y with 0:- Solve \[ -x + 4(0) = 6 \]- This simplifies to \[ -x = 6 \]- Solving for x gives us \[ x = -6 \]. Thus, the x-intercept is (-6, 0).Understanding this point is important since it indicates where the graph cuts the x-axis.
Identifying the Y-Intercept
The y-intercept is equally important for understanding the shape and position of a graph. It is the point where the graph crosses the y-axis. To find this, we set x to zero and solve for y.

Taking our example equation: \[ -x + 4y = 6 \]we find the y-intercept by substituting x with 0:- Solve \[ 0 + 4y = 6 \]- This simplifies to \[ 4y = 6 \]- Solve for y, leading to \[ y = \frac{3}{2} \]- Therefore, the y-intercept is (0, 1.5).The y-intercept helps to locate where the line meets the y-axis on the graph.
Steps for Graphing a Linear Equation
Graphing a linear equation involves plotting points and drawing a line through them. Understanding the graph of a linear equation helps visualize solutions and function behavior.

Here's a step-by-step
  • Identify Intercepts: Start by finding both the x-intercept and y-intercept. These points are where the line crosses the axes and are crucial for understanding the graph's orientation.
  • Select a Checkpoint: A third point on the line helps ensure accuracy in plotting. Compute this by substituting another value for x or y, as was done with x = 1 in our example. This extra point aids in drawing a precise line.
  • Plot Points on the Graph: Use the calculated intercepts and the checkpoint to plot on the coordinate plane. Mark them clearly.
  • Draw the Line: Carefully connect these plotted points, extending the line beyond the given points to represent the equation fully. This line represents all solutions of the equation.
Graphing converts abstract equations into visual cues, easing comprehension.
Understanding the Coordinate Plane
A coordinate plane is a foundational concept in graphing linear equations. It's a two-dimensional surface used to plot points, lines, and curves.

Key features:
  • Axes: The plane consists of two perpendicular lines 鈥� the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin (0,0).
  • Quadrants: The plane is divided into four sections or quadrants, which help in identifying the position of points like intercepts.
  • Coordinates: Each point on the plane has a pair of numbers known as coordinates, with the first number indicating position along the x-axis and the second along the y-axis.
Understanding the coordinate plane helps in plotting accurate graphs for linear equations. It provides the framework for visual representation, making it easier to analyze and solve problems involving linear relationships.

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

The graph shows that in \(2000,45 \%\) of U.S. adults believed that most qualified students get to attend college. For the period from 2000 through 2010 , the percentage who believed that a college education is available to most qualified students decreased by approximately 1.7 each year. These \- conditions can be described by the mathematical model $$ Q=-1.7 n+45 $$ where \(Q\) is the percentage believing that a college \- education is available to most qualificd students \(n\) years after 2000 a. Let \(n=0,5,10,15,\) and \(20 .\) Make a table of values showing five solutions of the equation. b. Graph the formula in a rectangular coordinate system. Suggestion: Let each tick mark on the horizontal axis, labeled \(n\), represent 5 units. Extend the horizontal axis to include \(n=25 .\) Let each tick mark on the vertical axis, labeled \(Q\), represent 5 units and extend the axis to include \(Q=50\) \- c. Use your graph from part (b) to estimate the percentage of U.S. adults who will believe that a college education is available to most qualified students in 2018 . \- d. Use the formula to project the percentage of U.S. adults who will believe that a college education is available to most qualified students in 2018 .

Will help you prepare for the material covered in the first section of the next chapter. Is \((4,-1)\) a solution of both \(x+2 y=2\) and \(x-2 y=6 ?\)

Use a graphing utility to graph each equation. You will need to solve the equation for \(y\) before entering it. Use the graph displayed on the screen to identify the \(x\)-intercept and the \(y\)-intercept. \(2 x+y=4\)

Solve: \(\quad 8(x-2)-2(x-3) \leq 8 x\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of \(x
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Mechanics Maths

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.