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Chapter 2: Problem 34

Determine the slope and the \(y\) -intercept. $$ y=-x-6 $$

Short Answer

Expert verified
Slope: -1, y-intercept: -6

Step by step solution

01

Write the equation in slope-intercept form

The given equation is already in slope-intercept form, which is: \[ y = mx + b \]where \( m \) is the slope and \( b \) is the y-intercept.
02

Identify the slope

In the equation \( y = -x - 6 \), the coefficient of \( x \) is \( -1 \). Therefore, the slope is \( m = -1 \).
03

Identify the y-intercept

In the equation \( y = -x - 6 \), the constant term is \( -6 \). Therefore, the y-intercept is \( b = -6 \).

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Key Concepts

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

slope
The slope of a line is a measure of its steepness. It's found in the equation of the line written in slope-intercept form as \( y = mx + b \). In this equation, the slope is represented by the letter \( m \).

To determine the slope from an equation, simply look at the coefficient of the \( x \) term. For example, in the equation \( y = -x - 6 \), the slope is \( -1 \) because the coefficient of \( x \) is \( -1 \).

The slope tells you how much the \( y \)-value of a point on the line changes for a one-unit increase in the \( x \)-value. A positive slope means the line rises as you move from left to right, while a negative slope means it falls.

In summary, the slope:
  • Determines the steepness and direction of the line
  • Is represented by \( m \) in the equation \( y = mx + b \)
  • Is the coefficient of the \( x \) term.
y-intercept
The \( y \)-intercept is the point where the line crosses the \( y \)-axis. This happens when the \( x \)-value is zero. When a line's equation is in slope-intercept form, the \( y \)-intercept can be directly identified.

In the equation \( y = mx + b \), the \( y \)-intercept is represented by \( b \). This value tells you the starting point of the line on the \( y \)-axis.

Using our example equation \( y = -x - 6 \), the \( y \)-intercept is \( -6 \). This means that the line crosses the \( y \)-axis at the point (0, -6).

Key points about the \( y \)-intercept:
  • It's where the line intersects the \( y \)-axis.
  • Represented by \( b \) in the equation \( y = mx + b \).
  • Found when the \( x \)-value is zero.
linear equations
Linear equations describe a straight-line graph and are fundamental in algebra. They typically look like this: \( y = mx + b \), known as the slope-intercept form.

Linear equations are called 'linear' because they create straight lines when graphed on a coordinate plane. Each solution pair \( (x, y) \) of the equation lies on this line.

Some important points about linear equations include:
  • They always graph as straight lines.
  • The 'slope-intercept form' \( y = mx + b \) is very useful for quickly graphing and understanding these lines.
  • The slope \( m \) tells you how steep the line is.
  • The \( y \)-intercept \( b \) tells you where the line crosses the \( y \)-axis.
In our exercise equation \( y = -x - 6 \), you see that:
  • The equation is already in slope-intercept form.
  • The slope (\( m = -1 \)) shows the line slopes downwards.
  • The \( y \)-intercept (\( b = -6 \)) shows the line crosses the \( y \)-axis at -6.
By understanding these components, you can easily master the basics of linear equations and their graphs.

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