/*! 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 82 Explain how to graph \(A x+B y=C... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 82

Explain how to graph \(A x+B y=C\) if \(C\) is not equal to zero.

Short Answer

Expert verified
The graph of \(Ax + By = C\) can be created by first isolating y, which results in \(y = -\frac{A}{B}x + \frac{C}{B}\). Then identify the slope and y-intercept, which are \(-\frac{A}{B}\) and \(\frac{C}{B}\) respectively, then plot the y-intercept on the graph and use the slope to find another point. Connecting the two points gives the graph of the equation.

Step by step solution

01

Write the equation

We start with the equation \(Ax + By = C\).
02

Isolate y in terms of x

To isolate y, first subtract \(Ax\) from both sides, that result in \(By = -Ax + C\). Then, divide every term by \(B\), so the equation becomes \(y = -\frac{A}{B}x + \frac{C}{B}\)
03

Identify the slope and y-intercept

The equation is now in the form y = mx + b. The coefficient of x is the slope of the line, and the constant term is the y-intercept. So, the slope, \(m\), is \( -\frac{A}{B}\) and the y-intercept, \(b\), is \(\frac{C}{B}\).
04

Plot the graph

After identifying the slope (m) and the y-intercept (b), plot the y-intercept on the y-axis. Then, use the slope to find another point on the line. The slope tells us to move vertically -A unit(s) and horizontally B unit(s) from the y-intercept. Once we've plotted a second point, we can draw a straight line connecting the two points, which represents the graph of the given 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.

Slope-intercept form
The slope-intercept form is a way of expressing the equation of a straight line. It is written as \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept. This form is incredibly useful because it immediately reveals both the slope and the y-intercept, making graphing on a coordinate plane straightforward. Graphs can be quickly sketched by starting at the y-intercept and using the slope to find other points.
Coordinate plane
A coordinate plane is a two-dimensional surface where we can graph lines and curves. It is made up of a horizontal axis, known as the x-axis, and a vertical axis, called the y-axis. These axes meet at a point called the origin. On this plane, each point has a pair of coordinates, \( (x, y) \), which specify its location relative to the axes. The coordinate plane allows us to visually represent equations like lines intersecting or parallel lines based on their equations.
Slope and y-intercept
In the equation \( y = mx + b \), the slope \( m \) defines the steepness and direction of the line. A positive slope means the line rises from left to right, while a negative slope means it falls. The value of \( m \) is calculated as the change in \( y \) divided by the change in \( x \) between any two points on the line. The y-intercept \( b \) is where the line crosses the y-axis, and it tells us the value of \( y \) when \( x = 0 \). Knowing both slope and y-intercept is essential for accurately graphing linear equations on the coordinate plane.
Linear function graphing
Graphing a linear function involves plotting its equation on the coordinate plane. Start with the y-intercept \( b \), which is the point where the line meets the y-axis. From this point, use the slope \( m \) to find other points. If \( m = \frac{3}{2} \), for example, move 3 units up for each 2 units right from the y-intercept to mark the next point. Connect these points with a straight line, extending infinitely in both directions. Graphing helps to visualize the relationship between variables represented by the linear equation.

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

Find the absolute value: \(|-13.4|\) (Section \(1.3,\) Example 8 )

If you are given an equation of the form \(A x+B y=C\) explain how to find the \(y\) -intercept.

A formula in the form \(y=m x+b\) models the cost, \(y,\) of a four-year college \(x\) years after \(2010 .\) Would you expect \(m\) to be positive, negative, or zero? Explain your answer.

Solve: \(\quad 8(x-2)-2(x-3) \leq 8 x\) (Section \(2.7,\) Example 8 )

Make Sense? In Exercises \(70-73\), determine whether each statement "makes sense" or "does not make sense" and explair your reasoning. The slope-intercept form of a line's equation makes it possible for me to determine immediately the slope and the \(y\) -intercept.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete 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.