/*! 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 19 Find the \(x\) - and \(y\) -inte... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 19

Find the \(x\) - and \(y\) -intercepts. $$3 x-y=-12$$

Short Answer

Expert verified
The x-intercept is at the point (-4, 0), and the y-intercept is at the point (0, 12).

Step by step solution

01

Find the x-intercept

To find the x-intercept, \(y\) is set equal to zero, and \(x\) is solved for. So in the equation \(3x - y = -12\), substitute \(y = 0\). This will change the equation to \(3x - 0 = -12\), which simplifies to \(3x = -12\). Solving for \(x\) by dividing both sides by 3 results in: \(x = -12 / 3 = -4\). So the x-intercept is at the point (-4, 0).
02

Find the y-intercept

To find the y-intercept, \(x\) is set equal to zero, and \(y\) is solved for. Substitute \(x = 0\) into the original equation \(3x - y = -12\) to get \(3(0) - y = -12\). This simplifies to \(-y = -12\). Solving for \(y\) by multiplying both sides by -1, results in \(y = 12\). So the y-intercept is at the point (0, 12).

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 line or curve is the point where it crosses the x-axis. At this point, the y-coordinate is always zero. Understanding how to find the x-intercept is crucial for analyzing linear equations and graphing them on a coordinate plane. Here's how it's done:
  • Set \(y\) equal to zero: Since the intercept lies on the x-axis, the value of \(y\) is 0.
  • Solve for \(x\): Substitute \(y = 0\) into the equation and solve for \(x\). For example, in the equation \(3x - y = -12\), substituting \(y = 0\) gives \(3x = -12\). Dividing both sides by 3 results in \(x = -4\).
  • Identify the intercept point: The x-intercept is written as a coordinate point: \((-4, 0)\) in this case.
Finding the x-intercept helps to visualize the behavior of the line in relation to the x-axis. This is one of the first steps in analyzing any linear equation. By locating the intercept, you set the stage for graphing the entire line.
Y-Intercept
The y-intercept is found where a line crosses the y-axis. At this point, the x-coordinate is always zero. Learning how to determine this intercept is essential for graphing linear equations accurately. Here鈥檚 a simple way to find the y-intercept:
  • Set \(x\) to zero: Because the intercept is located on the y-axis, \(x\) must be 0.
  • Solve for \(y\): Substitute \(x = 0\) into the equation and solve for \(y\). For example, with \(3x - y = -12\), substituting \(x = 0\) results in \(-y = -12\). Multiplying both sides by -1 gives \(y = 12\).
  • Identify the intercept point: The y-intercept is then expressed as a coordinate point: \((0, 12)\) in this example.
The y-intercept gives you a starting point on the graph, marking where the line meets the y-axis. This information, combined with the x-intercept, offers a clear picture of how the line behaves on a coordinate system.
Coordinate System
The coordinate system is a framework used to locate points on a plane. By understanding this system, you can interpret and plot linear equations effectively. Here's a breakdown of the core principles:
  • Axes: There are two axes 鈥� the horizontal x-axis and the vertical y-axis. These intersect at the origin, \((0, 0)\).
  • Quadrants: The coordinate plane is divided into four quadrants, each representing a combination of positive and negative values of \(x\) and \(y\).
  • Plotting Points: Any point is defined by an ordered pair \((x, y)\). For example, the point \((3, -2)\) lies 3 units to the right of the origin and 2 units down.
Using a coordinate system, you can graph lines and curves by plotting points determined by intercepts and other solutions to equations. It helps visualize mathematical relationships on a plane, providing a graphical context to numerical data. This forms the basis of many geometric and algebraic concepts in mathematics.

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

graph \(y=x+3, y=2 x+3,\) and \(y=-\frac{1}{2} x+3 .\) What observation can you make about the graphs?

Evaluate a function. Given \(f(x)=5 x+1,\) find \(f(2)\).

Dis Suppose \(A, B,\) and \(C\) are all positive numbers. Does the \(y\) -intercept of the graph of \(A x+B y=C\) lie above or below the \(x\) -axis? Does the graph slant upward to the right or downward to the right?

Determine whether there is a line that contains all of the given points. If so, find the equation of the line. $$(3,-1),(12,-4),(-6,2)$$

Evaluate a function. Given \(Q(r)=4 r^{2}-r-3,\) find \(Q(2)\).
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.