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Chapter 3: Problem 39

Write an equation in slope-intercept form for the line that satisfies the given conditions. (Lesson \(3-4\) ) \(m=0.3, y\) -intercept is \(-6\)

Short Answer

Expert verified
The equation is \( y = 0.3x - 6 \).

Step by step solution

01

Understanding the Slope-Intercept Form

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

Substitute the Slope Value

We are given that the slope \( m \) of the line is 0.3. Substitute 0.3 for \( m \) in the slope-intercept equation: \( y = 0.3x + b \).
03

Substitute the Y-Intercept Value

We are informed that the y-intercept \( b \) is -6. Replace \( b \) with -6 in the equation: \( y = 0.3x - 6 \).
04

Finalize the Equation

After substituting both the values for \( m \) and \( b \), the equation of the line in slope-intercept form is \( y = 0.3x - 6 \).

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Key Concepts

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

Linear Equations
A linear equation is one of the simplest forms of mathematical expressions to understand. These equations are used to describe straight lines on a graph, and they typically look like this:
  • Linear equations indicate a direct relationship between the x-values and y-values. As x increases, y changes at a constant rate.
  • They always create a straight line when graphed, hence their name "linear."
Linear equations take a standard form, often seen as either the slope-intercept form or the standard form. For instance, in slope-intercept form, we have the equation represented as \( y = mx + b \). Here,
  • \( m \) stands for the slope, which dictates the "steepness" or inclination of the line.
  • \( b \) is the y-intercept, indicating the point where the line crosses the y-axis.
As linear equations only involve one power of the variable (x), they are often our first introduction to the world of algebraic expressions and can also be utilized to model real-life situations like budgeting or speed calculations.
Slope
The slope of a line is a critical element when working with linear equations.It describes the tilt or inclination of the line and is usually represented by the symbol \( m \). To further understand the slope, consider its impact:
  • A slope of 0 means the line is perfectly horizontal, no rise or fall.
  • If the slope is positive (as in our example where \( m = 0.3 \)), the line ascends from left to right. This demonstrates an increase in y when x increases.
  • A negative slope indicates a line that descends from left to right, showing a decrease in y as x increases.
The actual value of the slope quantifies how much y changes for a unit change in x. In mathematical terms, this is often expressed as "rise over run," which is calculated using the formula: \( m = \frac{\Delta y}{\Delta x} \), where \( \Delta \) denotes the change. Thus, in practical terms, the slope serves as a crucial determinant in predicting behavior patterns and trends.
Y-Intercept
The y-intercept of a line is a special feature in linear equations, denoted by the letter \( b \) in the slope-intercept form \( y = mx + b \). Essentially, it's a pivot point:
  • The y-intercept is the y-coordinate where the line crosses the y-axis of the graph.
  • In understanding an equation, the y-intercept tells you the starting value on the y-axis, where the x value is zero.
For example, if the y-intercept is \( -6 \), the line intersects the y-axis at the point \( (0, -6) \). This number can have practical implications too:
  • It might represent a baseline value, such as the initial cost or starting amount before comparisons come into play.
  • In financial terms, \( b \) might signify a fixed cost or a starting point in a growth model.
Moreover, solving for the y-intercept is usually straightforward, but understanding its implications can offer valuable insights when assessing different situations depicted by the equation. By knowing where the line starts on the y-axis, it provides a complete picture alongside the slope to understand a line's behavior effectively.

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Most popular questions from this chapter

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