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Chapter 1: Problem 17

Find an equation of the given line. \(x\)-intercept is \(-\pi ; y\)-intercept is 1

Short Answer

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

Step by step solution

01

Understand the intercepts

The given information states that the x-intercept of the line is \(-\pi\) and the y-intercept is 1. The intercept form of the equation of a line is \(\frac{x}{a} + \frac{y}{b} = 1\), where \(a\) is the x-intercept and \(b\) is the y-intercept.
02

Substitute the intercepts into the intercept form

Using the intercept form \(\frac{x}{a} + \frac{y}{b} = 1\) and substituting \(a = -\pi\) and \(b = 1\), the equation becomes \(\frac{x}{-\pi} + \frac{y}{1} = 1\).
03

Simplify the equation

To simplify, we rearrange the equation \(\frac{x}{-\pi} + y = 1\). This can be further multiplied by \-\pi\ to clear the fraction, yielding \(x - \pi y = -\pi\).

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Key Concepts

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

x-intercept
The x-intercept of a line is the point where the line crosses the x-axis. This point tells us where the y-value is zero.
To find the x-intercept, set the y-variable to zero in the line's equation and solve for x. For example, if an equation of a line is given as \( y = -0.5x + 3 \), to find the x-intercept, do the following:
  • Set y to 0: \( 0 = -0.5x + 3 \)
  • Solve for x: \( x = 6 \)
The x-intercept in this case is (6,0). In our main problem, the x-intercept is given as \( -\pi \).
y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. This point indicates where the x-value is zero.
Finding the y-intercept requires setting the x-variable to zero and solving for y. For an equation like this: \( y = 2x + 5 \), to determine the y-intercept, follow these steps:
  • Set x to 0: \( y = 2(0) + 5 \)
  • Simplify to find y: \( y = 5 \)
Thus, the y-intercept is (0, 5). In our example problem, the y-intercept is 1, which makes the intercept point (0,1).
intercept form
Intercept form of a line's equation is a way to write the equation using the x-intercept (a) and y-intercept (b).
This form is convenient because it directly uses the intercept points without needing to convert first. The equation in intercept form is written as: \( \frac{x}{a} + \frac{y}{b} = 1 \)
Here, \( a \) represents the x-intercept and \( b \) represents the y-intercept.
In our problem, substituting the x-intercept \( a = -\pi \) and the y-intercept \( b = 1 \), the intercept form becomes: \( \frac{x}{-\pi} + \frac{y}{1} = 1 \).
simplifying equations
Simplifying an equation involves making it easier to read and understand without changing its meaning.
It typically means eliminating fractions, combining like terms, and making the expression more concise.
For the intercept form equation \( \frac{x}{-\pi} + \frac{y}{1} = 1 \), we simplify by eliminating the fraction:
  • Multiply the entire equation by \( -\pi \).
  • This changes it to: \( x - \pi y = -\pi \).
The simplified intercept form equation is easier to interpret and use in further calculations.

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Most popular questions from this chapter

Using the sum rule and the constant-multiple rule, show that for any functions \(f(x)\) and \(g(x).\) $$\frac{d}{d x}[f(x)-g(x)]=\frac{d}{d x} f(x)-\frac{d}{d x} g(x).$$

Use limits to compute the following derivatives. $$f^{\prime}(0), \text { where } f(x)=x^{2}+2 x+2$$

Compute the following limits. $$\lim _{x \rightarrow-\infty} \frac{1}{x^{2}}$$

Suppose that 5 mg of a drug is injected into the bloodstream. Let \(f(t)\) be the amount present in the bloodstream after \(t\) hours. Interpret \(f(3)=2\) and \(f^{\prime}(3)=-.5 .\) Estimate the number of milligrams of the drug in the bloodstream after \(3 \frac{1}{2}\) hours.

Compute the following limits. $$\lim _{x \rightarrow \infty} \frac{1}{x^{2}}$$
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