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Chapter 5: Problem 102

What is the \(y\) -intercept of the graph of \(y=-3 x-5 ?\)

Short Answer

Expert verified
The y-intercept is -5.

Step by step solution

01

Identify the Linear Equation Structure

The given equation is in the slope-intercept form, which is generally expressed as \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
02

Locate the Y-Intercept in the Equation

In the equation \( y = -3x - 5 \), the term \(-5\) represents the y-intercept. Therefore, the y-intercept is \( c = -5 \).
03

Conclude the Y-Intercept Value

Since the y-intercept is the point where the graph crosses the y-axis, our calculation shows that this occurs at \( (0, -5) \).

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Key Concepts

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

Linear Equation
A linear equation is an essential concept in algebra that represents a straight line when graphed on a coordinate plane. It is often described in the form of \( y = mx + c \), where \( m \) stands for the slope, \( x \) is the variable representing the horizontal axis, and \( c \) is the constant term known as the y-intercept. Linear equations form the foundation for many real-world problems, such as calculating budgets, predicting trends, or modeling physical phenomena.

Understanding linear equations allows us to:
  • Predict values for given variables
  • Understand the relationship between different quantities
  • Analyze and interpret data in various fields
Approaching linear equations involves recognizing their structure and predicting how changes in one quantity affect another.
Slope-Intercept Form
The slope-intercept form is a popular way to write linear equations. It is expressed as \( y = mx + c \). The components of this equation give us direct insight into the characteristics of the graph.

  • **Slope** \( (m) \): It indicates the steepness and direction of the line. A positive slope means the line goes upwards from left to right, while a negative slope points downward.
  • **Y-Intercept** \( (c) \): This is where the line crosses the y-axis. It tells us the value of \( y \) when \( x = 0 \).
Using the slope-intercept form makes it easy to graph a linear equation and quickly understand the relationship between \( x \) and \( y \). Knowing how to rewrite equations into this form is crucial because it simplifies solving systems of equations and interpreting line intersections.
Graphing Linear Equations
Graphing linear equations is a visual way to represent the relationships described by linear equations. Working with the slope-intercept form, the graphing process becomes straightforward and intuitive.

Here are the steps to graph a linear equation like \( y = -3x - 5 \):
  • Identify the y-intercept \( (0, c) \), in this case, \( (0, -5) \).
  • Use the slope \((m)\), which is \(-3\) here. This means for every increase of 1 in \( x \), \( y \) decreases by 3.
  • Starting from the y-intercept, apply the slope to determine another point on the line. For example, moving right by 1 on the x-axis means going down by 3 on the y-axis.
  • Draw the line through these points to complete the graph.
This approach helps in visually understanding how the line behaves and allows for the quick identification of the y-intercept, slope, and potential intersections with other lines.

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Most popular questions from this chapter

We can find \((2 x+3)^{2}\) and \((5 y-6)^{2}\) using the FOIL method or using special product rules. Explain why the special product rules are faster.

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