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Chapter 0: Problem 44

Write the equation of the line, given the slope and intercept. Slope: \(m=-2\) \(y\) -intercept: (0,1)

Short Answer

Expert verified
The equation of the line is \( y = -2x + 1 \).

Step by step solution

01

Understand the Line Equation

The general form of the linear equation in slope-intercept form is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. Our task is to plug in the given values into this equation.
02

Substitute Slope into the Equation

Substitute the given slope \( m = -2 \) into the equation. This changes the equation to \( y = -2x + b \).
03

Substitute the y-intercept into the Equation

The \( y \)-intercept is given as (0,1), meaning \( b = 1 \). Substitute \( b = 1 \) into the equation. This updates the equation to \( y = -2x + 1 \).
04

Combine into Final Equation

Now that we have both the slope and \( y \)-intercept in place, the final equation of the line is \( y = -2x + 1 \).

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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 mathematical expressions that represent straight lines in a coordinate plane. They come in several forms, but one of the most common is the slope-intercept form. This form of a linear equation is expressed as \( y = mx + b \). Here is how it breaks down:
  • \( y \) represents the dependent variable or the output of the equation.
  • \( x \) is the independent variable or the input.
  • \( m \) is the slope of the line, which indicates how steep the line is.
  • \( b \) is the y-intercept, which is where the line crosses the y-axis.
Linear equations are essential tools in algebra because they provide simple models for relationships and processes that change at a constant rate. You will often use them to predict values or analyze how different variables are interrelated.
The Slope: Rate of Change
The slope of a linear equation, represented by \( m \), is essential in determining the line's direction and steepness. It tells us how much the y-value changes for a unit change in the x-value. In simple terms, it is the rate at which one variable changes over another.
Imagine a hill:
  • If the slope is positive, like climbing up a hill, the line rises as you move to the right.
  • If the slope is negative, like climbing down a hill, the line falls as you move to the right.
  • A slope of zero means the line is flat, while an undefined slope (division by zero) represents a vertical line.
For example, in the equation \( y = -2x + 1 \), the slope \( m = -2 \) indicates that for each increase of 1 unit in \( x \), \( y \) decreases by 2 units. This negative slope results in a downward sloping line from left to right.
Identifying the Y-Intercept
The y-intercept of a linear equation is represented by \( b \), which is the point where the line crosses the y-axis. Knowing the y-intercept is crucial because it provides a starting value of the line when \( x = 0 \). You can locate this intercept by simply observing the equation in slope-intercept form.
For instance, in the equation \( y = -2x + 1 \), the y-intercept is 1. This means the line intersects the y-axis at the point \( (0,1) \).
  • This location gives you a specific starting point to measure the slope from, helping to graph the whole line with ease.
  • The y-intercept is often used in conjunction with the slope to graph a line quickly, starting at (0,\( b \)) and following the direction implied by the slope.
Recognizing the y-intercept lets you understand where the line stands in relation to the origin and is critical to drawing accurate graphs and making predictions based on the linear model.

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Most popular questions from this chapter

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