/*! 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 33 Find the \(x\) -intercept and th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 33

Find the \(x\) -intercept and the \(y\) -intercept of the line with the given equation. Sketch the line using the intercepts. A calculator can be used to check the graph. $$y=3 x+6$$

Short Answer

Expert verified
The x-intercept is \((-2, 0)\) and the y-intercept is \((0, 6)\).

Step by step solution

01

Identify the Equation Form

The given equation is in the slope-intercept form, which is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this equation, \(y = 3x + 6\), the slope \(m = 3\) and the y-intercept \(b = 6\).
02

Find the Y-Intercept

To find the y-intercept, set \(x = 0\) in the equation. Substitute \(x = 0\) into \(y = 3x + 6\): \[ y = 3(0) + 6 = 6 \]So, the y-intercept is the point \((0, 6)\).
03

Find the X-Intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\). Substitute \(y = 0\) into the equation:\[ 0 = 3x + 6 \]Subtract \(6\) from both sides:\[-6 = 3x \]Divide both sides by \(3\):\[ x = -2 \]The x-intercept is the point \((-2, 0)\).
04

Sketch the Line Using the Intercepts

Plot the intercepts on a coordinate plane: the y-intercept \((0, 6)\) and the x-intercept \((-2, 0)\). Draw a straight line through these two points, which extends in both directions. This line represents the graph of \(y = 3x + 6\).
05

Verify with a Calculator

Using a graphing calculator, enter the equation \(y = 3x + 6\) to display the graph. Ensure that the intercepts match the points \((0, 6)\) and \((-2, 0)\) that were calculated.

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
Linear equations often appear in what is known as the slope-intercept form: \( y = mx + b \). This format allows us to quickly determine essential information about a line on a graph. In this equation:
  • \( m \) represents the slope of the line, indicating how steep the line is and the direction it slants. If \( m \) is positive, the line increases; if negative, it decreases.
  • \( b \) is the y-intercept. This is where the line crosses the y-axis.
For the equation \( y = 3x + 6 \), the slope \( m = 3 \), so the line rises 3 units up for every unit it moves to the right. The y-intercept here is 6, meaning the line crosses the y-axis at the point (0, 6). This structure makes it simple to sketch lines even without complex algebra.
X-Intercept
The x-intercept is the point where the line crosses the x-axis. At this point, the value of \( y \) is zero because the line has not moved up or down from this axis. To find the x-intercept of a line, we set \( y = 0 \) in the equation and solve for \( x \). For the equation \( y = 3x + 6 \):
  • Set \( y = 0 \) which gives \( 0 = 3x + 6 \).
  • Subtract 6 from both sides to get \( -6 = 3x \).
  • Divide both sides by 3 to find \( x = -2 \).
Thus, the x-intercept is the point \( (-2, 0) \). At this coordinate, the line touches the x-axis, making it a crucial element for graphing a line.
Y-Intercept
Unlike the x-intercept, the y-intercept is where the line crosses the y-axis. At this point, \( x \) is zero because the line hasn't moved left or right from the axis. To identify the y-intercept, simply substitute \( x = 0 \) into the equation and solve for \( y \). In our example equation \( y = 3x + 6 \):
  • Enter \( x = 0 \), resulting in the expression \( y = 3(0) + 6 \).
  • Calculate to find \( y = 6 \).
Therefore, the y-intercept of the line is at \( (0, 6) \). This intercept is essential for understanding where the line starts on a graph in the vertical direction.
Graphing a Line
Graphing a line using its intercepts is a straightforward yet effective method. A line is fully defined on a graph once two points are known. Here's how to sketch a line using intercepts:
  • Begin by plotting the y-intercept. In our example, start at point \( (0, 6) \) on the graph.
  • Plot the x-intercept, which for \( y = 3x + 6 \) is \( (-2, 0) \).
  • Draw a straight line through these two plotted points, making sure to extend the line in both directions beyond the intercepts.
When finished, the line visually represents the equation \( y = 3x + 6 \). Checking with a graphing calculator can assure accuracy. You should see a line crossing the y-axis at 6 and the x-axis at \( -2 \). This visualization helps reinforce understanding of slope and intercept concepts.

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

Set up appropriate systems of two linear equations and solve the systems algebraically. All data are accurate to at least two significant digits. In a test of a heat-seeking rocket, a first rocket is launched at \(2000 \mathrm{ft} / \mathrm{s},\) and the heat-seeking rocket is launched along the same flight path 12 s later at a speed of \(3200 \mathrm{ft} / \mathrm{s}\). Find the times \(t_{1}\) and \(t_{2}\) of flight of the rockets until the heat- seeking rocket destroys the first rocket.

Evaluate the given third-order determinants. $$\left|\begin{array}{rrr} 0.25 & -0.54 & -0.42 \\ 1.20 & 0.35 & 0.28 \\ -0.50 & 0.12 & -0.44 \end{array}\right|$$

Show that the given systems of equations have either an unlimited number of solutions or no solution. If there is an unlimited number of solutions, find one of them. $$\begin{aligned}&x-2 y-3 z=2\\\&x-4 y-13 z=14\\\&-3 x+5 y+4 z=2\end{aligned}$$

Set up appropriate systems of two linear equations and solve the systems algebraically. All data are accurate to at least two significant digits. A person's email for a day contained a total of 78 messages. The number of spam messages was two less than four times the other messages. How many were spam?

Show that the given systems of equations have either an unlimited number of solutions or no solution. If there is an unlimited number of solutions, find one of them. $$\begin{aligned}&3 x+3 y-2 z=2\\\&2 x-y+z=1\\\&x-5 y+4 z=-3\end{aligned}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.