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Chapter 1: Problem 1

Find the slopes and \(y\) -intercepts of the following lines. \(y=3-7 x\)

Short Answer

Expert verified
The slope is \(-7\) and the y-intercept is \(3\).

Step by step solution

01

- Understand the Slope-Intercept Form

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

- Identify the Slope (\(m\))

Compare the given equation \(y = 3 - 7x\) with the slope-intercept form \(y = mx + b\). Here, we can rewrite the equation as \(y = -7x + 3\). Hence, the slope \(m\) is \(-7\).
03

- Identify the Y-Intercept (\(b\))

From the equation \(y = -7x + 3\), we see that the y-intercept \(b\) is \(3\).

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Key Concepts

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

Linear Equations
Linear equations are equations that create straight lines when graphed on a coordinate plane. They follow a consistent relationship between the variables. The general format for these equations is known as the slope-intercept form, represented as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Understanding linear equations is essential because they help explain relationships and trends in various fields like economics, physics, and everyday life. By mastering linear equations, you can predict outcomes and understand changes over a range of variables.
Slope
The slope of a line measures its steepness and direction. It is represented by the letter \(m\) in the slope-intercept form of a linear equation. Essentially, the slope is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. Mathematically, it is calculated as:
\[ m = \frac{\Delta y}{\Delta x} \]
where \(\Delta y\) is the change in y-coordinates and \(\Delta x\) is the change in x-coordinates.
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope means the line falls as it moves from left to right.
  • A zero slope indicates a horizontal line.
  • An undefined slope signifies a vertical line.
For the given equation \[ y = -7x + 3 \], the slope \(m\) is \(-7\), indicating the line falls as it moves from left to right.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form \(y = mx + b\), the y-intercept is represented by the letter \(b\). This value indicates what y equals when x is zero, essentially showing where the line meets the y-axis.
To find the y-intercept in an equation, you can let \(x = 0\) and solve for \(y\). For the equation \[ y = -7x + 3 \], if you substitute \(x\) with \(0\):
\[ y = -7(0) + 3 \] Thus, \(y = 3\) means the line crosses the y-axis at (0, 3).
Equation Manipulation
Equation manipulation involves rewriting or rearranging an equation to make it easier to work with or understand. This skill is essential when handling linear equations to identify key components like the slope and y-intercept.
For example, given the equation \[ y = 3 - 7x \], we want it in the form \(y = mx + b\). By rearranging the terms, we get:
\[ y = -7x + 3 \] Through such manipulation:
  • \

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Most popular questions from this chapter

Find the indicated derivative. \(\frac{d}{d x}\left(x^{3 / 4}\right)\)

Use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. \(f^{\prime}(1)\), where \(f(x)=\frac{1}{1+x^{2}}\)

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Determine whether each of the following functions is continuous and/or differentiable at \(x=1\). \(f(x)=\left\\{\begin{array}{ll}\frac{1}{x-1} & \text { for } x \neq 1 \\ 0 & \text { for } x=1\end{array}\right.\)

Apply the three-step method to compute the derivative of the given function. \(f(x)=3 x^{2}-2\)
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