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Chapter 3: Problem 3

Write an equation in slope-intercept form of the line having the given slope and \(y\) -intercept. \(m=-\frac{3}{5}\) \(y\) -intercept at \((0,-2)\)

Short Answer

Expert verified
The equation in slope-intercept form is \( y = -\frac{3}{5}x - 2 \).

Step by step solution

01

Understand the Slope-Intercept Form Equation

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

Identify the Given Values

From the problem, identify the slope \( m = -\frac{3}{5} \) and the y-intercept, which is the point \((0, -2)\). This tells us \( b = -2 \).
03

Substitute the Values into the Equation

Plug the identified values into the slope-intercept form: \( y = mx + b \). Substitute \( m = -\frac{3}{5} \) and \( b = -2 \): \( y = -\frac{3}{5}x - 2 \).
04

Finalize and Verify the Equation

The equation of the line in slope-intercept form is: \( y = -\frac{3}{5}x - 2 \). You can verify this by ensuring that when \( x = 0\), \( y = -2 \) matches the y-intercept given.

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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 describe a straight line on a coordinate plane. They follow the general form of \( y = mx + b \), where \( y \) and \( x \) are variables that represent coordinates on the graph. This equation showcases how \( y \) depends on \( x \), with the slope \( m \) indicating how steep the line is and \( b \) determining where the line crosses the y-axis. Key characteristics of linear equations include:
  • A constant rate of change represented by the slope.
  • A straight-line graph that extends infinitely in both directions.
  • Each solution is a unique coordinate pair \((x, y)\) that lies on the line.
These equations are incredibly useful for modeling real-world situations where there's a constant change or relationship between two variables.
Grasping the Concept of Slope
The slope of a line offers a measure of its steepness and direction. In the equation \( y = mx + b \), \( m \) is the slope. Mathematically, slope is the ratio of the rise (change in \( y \)) to the run (change in \( x \)). It is usually expressed as a fraction or a decimal:\[ m = \frac{\text{rise}}{\text{run}} \]Important aspects of slope:
  • A positive slope indicates the line is rising as it moves from left to right.
  • A negative slope suggests the line is falling as it moves from left to right, as in the example \( m = -\frac{3}{5} \).
  • A slope of zero indicates a horizontal line, where there is no rise, only a run.
  • An undefined slope corresponds to a vertical line.
Understanding the slope is crucial for predicting how changes in one variable affect another within linear relationships.
Defining the Y-Intercept
The y-intercept is the point where a line crosses the y-axis of a graph. In the equation \( y = mx + b \), the y-intercept is represented by \( b \). This point is where \( x = 0 \). It's crucial because it provides a starting point for graphing the line and gives insight into the line's position on the coordinate plane.For example, if you have a line with the equation \( y = -\frac{3}{5}x - 2 \), the y-intercept is \(-2\), meaning the line crosses the y-axis at \((0, -2)\).Key points to understand about y-intercepts:
  • It provides a specific starting point on the graph (when \( x = 0\)).
  • The intercept signifies the value of \( y \) independently of \( x \).
  • It allows us to quickly sketch the line on a graph alongside the slope.
By understanding the y-intercept, you can better visualize and plot linear equations in slope-intercept form.

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