/*! 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 Find an equation of the line tha... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 25

Find an equation of the line that satisfies the given conditions. Slope \(3 ; \quad y\) -intercept \(-2\)

Short Answer

Expert verified
The equation of the line is \( y = 3x - 2 \).

Step by step solution

01

Recall the Slope-Intercept Form

The equation of a line in slope-intercept form is given by \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept.
02

Substitute the Given Slope

Substitute the given slope \( m = 3 \) into the slope-intercept formula. This gives us \( y = 3x + b \).
03

Substitute the Given Y-Intercept

Now, substitute the given y-intercept \( b = -2 \) into the equation from Step 2. This yields \( y = 3x - 2 \).

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.

Linear Equations
Linear equations are mathematical expressions that describe a straight line when graphed. They have a constant rate of change. This means every time the value of the variable changes, the graph's rise to run ratio remains constant. A linear equation can come in different forms, but one commonly used format is the slope-intercept form:
  • The standard form is given as: \( Ax + By = C \).
  • The slope-intercept form: \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept.
  • Each form of a linear equation tells us different things about the line, such as its slope, y-intercept, and direction.
Linear equations are central to algebra and are fundamental to understanding concepts in more advanced mathematics. With these equations, you can model and solve real-world problems that require constant change or growth.
Slope
The slope of a line is a measure of its steepness and direction. It tells us how many units the line rises or falls for a unit increment horizontally. The slope is represented by \( m \) in the equation \( y = mx + b \). It is defined mathematically as:
  • \( m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} \), which means the change in \( y \) over the change in \( x \).
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope indicates the line falls as it moves from left to right.
  • If the slope is zero, the line is horizontal, indicating no vertical change.
  • A vertical line has an undefined slope because division by zero is not possible.
Understanding the slope is crucial, as it determines how "tilted" a line is on a graph, and provides information about the rate of change between the x and y variables.
Y-Intercept
The y-intercept is the point where a line crosses the y-axis. In the slope-intercept form equation \( y = mx + b \), \( b \) represents the y-intercept. This point shows the value of \( y \) when \( x = 0 \). Here's what we need to know about the y-intercept:
  • It's crucial for graphing a line because it provides a starting point.
  • For the given equation from the exercise, the y-intercept is \(-2\), meaning when \( x = 0 \), \( y = -2 \).
  • Every line has exactly one y-intercept unless it's vertical, in which case the line may not cross the y-axis.
The y-intercept helps us understand the starting value of a function, and it can be useful in determining where a line is positioned on a graph. Knowing \( b \) allows us to plot the line quickly and understand the behavior of linear functions at the origin.

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

Perform the indicated operations and simplify. $$\left(x^{2}-a^{2}\right)\left(x^{2}+a^{2}\right)$$

Factor the expression completely. $$2 x^{2}+5 x+3$$

Factor the trinomial. $$2(a+b)^{2}+5(a+b)-3$$

Factor the expression by grouping terms. $$x^{5}+x^{4}+x+1$$

(a) Show that \(a b=\frac{1}{2}\left[(a+b)^{2}-\left(a^{2}+b^{2}\right)\right]\) (b) Show that \(\left(a^{2}+b^{2}\right)^{2}-\left(a^{2}-b^{2}\right)^{2}=4 a^{2} b^{2}\) (c) Show that \(\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)=(a c+b d)^{2}+(a d-b c)^{2}\) (d) Factor completely: \(4 a^{2} c^{2}-\left(a^{2}-b^{2}+c^{2}\right)^{2}\)
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.