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Chapter 3: Problem 76

Find the slope of each line. The line with equation \(y=-3 x+4\)

Short Answer

Expert verified
The slope of the line is -3.

Step by step solution

01

Identify the Slope-Intercept Form

Recognize that the given equation \(y = -3x + 4\) is in the slope-intercept form \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.
02

Extract the Slope

In the equation \(y = -3x + 4\), compare it to the general form \(y = mx + b\). Here, \(m = -3\), indicating the slope of the line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Linear Equations
Linear equations are fundamental in algebra. They describe a straight line on a graph. Such equations measure relationships between two variables, typically expressed in the form of \(y = mx + b\).
Here, \(m\) is the slope and \(b\) is the y-intercept.
  • The slope \(m\) signifies the tilt or steepness of the line.
  • The y-intercept \(b\) is where the line crosses the y-axis.
Linear equations are easy to work with since they create a predictable and uniform graph. Understanding them is essential, as they are the foundation of many mathematical concepts beyond basic algebra.
Exploring the Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most practical and commonly used formats in algebra. It looks like \(y = mx + b\).
In this format, the slope \(m\) tells you how quickly the line rises or falls as you move along it, while the y-intercept \(b\) tells you the exact point where the line crosses the y-axis.
  • When \(m > 0\), the line ascends as it moves from left to right.
  • When \(m < 0\), the line descends.
  • If \(m = 0\), you have a horizontal line.
The slope-intercept form allows for quick identification and graphing of a line, making it a key tool in both simple and complex algebraic problems.
Your Guide to Algebra Tutorials
Algebra tutorials often start with linear equations and the slope-intercept form because they are straightforward and efficient.
These tutorials break down algebra into bite-sized lessons to ease learning.
To effectively understand and use algebra tutorials involving linear equations:
  • Follow each step and note how the equation is transformed.
  • Make sure to practice by solving similar problems.
  • Fully grasp terms like slope and y-intercept for greater success.
These guides simplify algebraic concepts and help strengthen your foundational understanding, preparing you for more advanced topics in mathematics.

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Most popular questions from this chapter

\(X\) -Files, Will \& Grace, and Seinfeld are three of the most popular television shows of all time. The total number of episodes of these three shows is \(575 .\) There are 21 more episodes of \(X\) -Files than Seinfeld , and the difference between the number of episodes of Will \& Grace and Seinfeld is 14 Find the number of episodes of each show.

Use Cramer's rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} 4 x-3 y=1 \\ 6 x-8 z=1 \\ 2 y-4 z=0 \end{array}\right. $$

Solve: \(\left\\{\begin{array}{l}2 a b-3 c d=1 \\ 3 a b-2 c d=1\end{array}\right.\) for \(a\) and \(c .\) Assume that \(b\) and \(d\) are nonzero constants.

For the function \(y=2 x^{2}+6 x+1,\) what is the independent variable and what is the dependent variable?

Use Cramer's rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} x+y+2 z=7 \\ x+2 y+z=8 \\ 2 x+y+z=9 \end{array}\right. $$
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