/*! 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 32 Graph each linear equation. \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 32

Graph each linear equation. \(x=-y+6\)

Short Answer

Expert verified
Plot (0,6), use slope -1 to plot (1,5), draw line through points.

Step by step solution

01

Rewrite the Equation

Rewrite the given equation in the slope-intercept form, which is \(y = mx + b\). In this case, starting with the equation \(x = -y + 6\), solve for \(y\) to get: \[y = -x + 6.\]
02

Identify the Slope and Intercept

From the slope-intercept form \(y = mx + b\), we can identify the slope (m) and the y-intercept (b). Here, the slope (m) is -1 and the y-intercept (b) is 6.
03

Plot the Y-Intercept

Start by plotting the y-intercept on the graph. The y-intercept is 6, so place a point at (0, 6) on the y-axis.
04

Use the Slope to Find Another Point

The slope is -1, which means for every 1 unit you move to the right along the x-axis, you move 1 unit downwards along the y-axis. Starting from the y-intercept (0, 6), move 1 unit to the right to x = 1 and 1 unit down to y = 5, placing another point at (1, 5).
05

Draw the Line

Draw a straight line through the points (0, 6) and (1, 5). This line represents the graph of the equation \(x = -y + 6\).

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 one of the most common ways to write the equation of a straight line. It is represented as \( y = mx + b \). Here:
  • \( m \) is the slope of the line,
  • \( b \) is the y-intercept.
This form makes it easy to graph a line. You can immediately see the slope and where the line crosses the y-axis.In our exercise, we rewrote the given equation from \( x = -y + 6 \) to the slope-intercept form \( y = -x + 6 \). Now, we can easily identify the slope and the y-intercept.
y-intercept
The y-intercept is the point where a line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the y-intercept is given by \( b \).In our equation, \( y = -x + 6 \), the y-intercept is 6. This means the line crosses the y-axis at the point (0, 6).When graphing a linear equation, starting with the y-intercept helps establish the initial position of the line on the graph.
plotting points
Plotting points involves placing points on a graph based on the coordinates provided by the equation. Here鈥檚 how to do it for the equation \( y = -x + 6 \):
  • First, plot the y-intercept: This is where \( x = 0 \), and in our case, \( y = 6 \). Place a point at (0, 6).
  • Next, use the slope to find another point: Move 1 unit to the right along the x-axis, then move 1 unit downward along the y-axis according to the slope of \( -1 \). The second point will be (1, 5).
With these two points, you can now draw the line that represents the equation.
slope
The slope is a measure of how steep a line is. It is denoted by \( m \) in the slope-intercept form of the equation \( y = mx + b \). The slope is the ratio of the change in the y-coordinate to the change in the x-coordinate, often described as 'rise over run'.In our example, \( y = -x + 6 \), the slope \( m \) is \( -1 \). This means for every 1 unit you move to the right along the x-axis, you move 1 unit downwards along the y-axis.Understanding the slope helps you to determine the direction and steepness of the line on the graph.

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

For each pair of equations, give the slopes of the lines and then determine whether the two lines are parallel, perpendicular, or neither. See Example \(6 .\) $$ \begin{aligned} &3 x-2 y=6\\\ &2 x+3 y=3 \end{aligned} $$

Do \((3,4)\) and \((4,3)\) correspond to the same point in the plane? Explain.

Find the \(x\) -intercept and the \(y\) -intercept for the graph of each equation. $$ y=2.5 $$

Describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ 3 y=-6 $$

Plot and label each point in a rectangular coordinate system. $$ (-3,5) $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Statistics

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.