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Chapter 1: Problem 8

Find an equation of the given line. Slope is \(2 ;(1,-2)\) on line

Short Answer

Expert verified
y = 2x - 4

Step by step solution

01

Understand the Problem

You need to find an equation of a line that passes through the point \(1,-2\) and has a slope of 2.
02

Use the Point-Slope Formula

The point-slope formula of a line is given by: \[ y - y_1 = m(x - x_1) \] where \(m\) is the slope, and \( (x_1, y_1) \) is a point on the line.
03

Substitute the Given Values

Substitute \(m = 2\), \( x_1 = 1\), and \( y_1 = -2\) into the point-slope formula: \[ y - (-2) = 2(x - 1) \]
04

Simplify the Equation

Simplify the equation to put it in slope-intercept form (\(y = mx + b\)): \[ y + 2 = 2(x - 1) \] \[ y + 2 = 2x - 2 \] \[ y = 2x - 4 \]

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Key Concepts

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

Understanding the Point-Slope Formula
The point-slope formula is a powerful tool for finding the equation of a line. It is particularly useful when you know one point on the line and the slope of the line. The formula is expressed as \( y - y_1 = m(x - x_1)\).
  • \( m \) is the slope of the line.
  • \( (x_1, y_1) \) is a specific point on the line.
To use this formula, simply substitute the slope (\( m \)) and the coordinates of the point (\( x_1, y_1 \)). Then, rearrange the equation to find the desired form of the line equation. This formula is straightforward and helps you quickly derive the line's equation without much hassle. Remember to always simplify your equation after substituting the values.
Converting to Slope-Intercept Form
The slope-intercept form is another popular way to represent the equation of a line. It simplifies to \( y = mx + b \). Here, \( m \) is the slope, and \( b \) is the y-intercept.
  • Slope (\( m \)): Indicates how steep the line is.
  • Y-intercept (\( b \)): The point where the line crosses the y-axis.
In the example problem, we first applied the point-slope formula: \[ y - (-2) = 2(x - 1) \] By simplifying, we got: \[ y + 2 = 2x - 2 \] and then: \[ y = 2x - 4 \] This final form \( y = 2x - 4 \) is the slope-intercept form of the line. It visibly shows the slope (\( 2 \)) and the y-intercept (\( -4 \)). Converting to this form makes it easier to graph or understand the line's behavior.
Basics of Linear Equations
Linear equations are equations that represent a straight line when plotted on a graph. These equations have no exponents higher than one and can be written in various forms, including point-slope and slope-intercept forms.
  • They follow the format \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.
  • Linear equations are widely used in algebra, physics, engineering, and various other fields.
When solving an exercise involving linear equations, it's essential to recognize the line's slope and specific points through which it passes. With the given slope and point in our example, we used formulas to derive the line's full equation: \( y = 2x - 4\). Understanding the various forms and conversions among them is key to mastering linear equations and their applications in many real-world problems.

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Most popular questions from this chapter

Determine which of the following limits exist. Compute the limits that exist. $$\lim _{x \rightarrow 8} \frac{x^{2}+64}{x-8}$$

Compute the difference quotient $$\frac{f(x+h)-f(x)}{h}.$$ Simplify your answer as much as possible. $$f(x)=2 x^{3}+x^{2}$$

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If possible, define \(f(x)\) at the exceptional point in a way that makes \(f(x)\) continuous for all \(x.\) $$f(x)=\frac{(6+x)^{2}-36}{x}, x \neq 0$$

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