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Chapter 2: Problem 20

Find an equation of the line through (1,2) with the indicated slope. -2

Short Answer

Expert verified
The equation of the line is \( y = -2x + 4 \).

Step by step solution

01

Understand the point-slope form

To find the equation of a line with a given slope that passes through a specific point, we use the point-slope form of a linear equation: \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1)\) is the point on the line, and \(m\) is the slope.
02

Substitute the given values

In this exercise, the slope \(m\) is \(-2\), and the point \((x_1, y_1)\) is \((1, 2)\). Substitute these into the point-slope form: \( y - 2 = -2(x - 1) \).
03

Simplify the equation

Expand the right side of the equation: \( y - 2 = -2x + 2 \).
04

Rearrange into slope-intercept form

Add 2 to both sides to solve for \(y\): \( y = -2x + 2 + 2 \). Simplify to get: \( y = -2x + 4 \). This is the equation of the line in slope-intercept form \(y = mx + b\).

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Key Concepts

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

Point-Slope Form
The point-slope form is a foundational concept in understanding linear equations. This form is especially useful when you know a point through which a line passes and its slope. The general equation for the point-slope form is \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1)\) represents a specified point on the line, while \( m \) refers to the slope. For students working with this form, remember:
  • The slope \( m \) indicates the steepness and the direction of the line鈥攏egative means the line falls as you move from left to right, and positive means it rises.
  • The point \((x_1, y_1)\) is any point through which the line passes, often given in the problem.
This format is straightforward to apply; just insert the slope and point into the equation. Its simplicity makes it particularly handy when transitioning to other forms of linear equations, such as the slope-intercept form.
Slope-Intercept Form
The slope-intercept form is one of the most widely used formats for linear equations. This form is expressed as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This format is favorable because:
  • It clearly shows both the slope and the y-intercept, which are essential for graphing.
  • The y-intercept \( b \) is the value of \( y \) when \( x \) is zero. It tells you where the line crosses the y-axis.
To transition from point-slope form to slope-intercept form, you perform simple algebraic manipulations to isolate \( y \) on one side of the equation. This ability to easily shift forms makes the slope-intercept form versatile, useful for quickly visualizing and graphing lines.
Linear Equations
Linear equations form the cornerstone of algebra and are characterized by their straight-line graphs. These equations involve variables that have a maximal degree of one and can typically be written in various forms, including point-slope and slope-intercept. Here are a few key points:
  • Linear equations graph to straight lines, each uniquely defined by its slope and intercept.
  • The slope \( m \) measures the rate of change of the dependent variable (usually \( y \)) with respect to the independent variable (\( x \)).
  • Understanding the slope is crucial; it shows us how quickly values are increasing or decreasing.
  • The intercepts鈥攕olutions at \( x = 0 \) and \( y = 0 \)鈥攇ive vital data about line positioning on a graph.
Mastering linear equations is vital for progressing in mathematics and understanding real-world relationships and changes. Whether in physics, economics, or any data-driven field, linear models represent single-variable correlations that are foundational knowledge.

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Most popular questions from this chapter

The ceiling function \(g(x)=\lceil x]=\) is defined to be the least integer \(n\) that is greater than or equal to \(x\). Fill in the blanks. $$ g(x)=[x]=\left\\{\begin{array}{ll} \vdots & \\ \longrightarrow & -3

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