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Chapter 2: Problem 36

Determine the slope and the \(y\) -intercept. $$ y=4 x+9 $$

Short Answer

Expert verified
Slope: 4, y-intercept: 9

Step by step solution

01

Identify the general form of a linear equation

The given equation is in the form of a linear equation which can be generally written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Identify the slope

In the equation \( y = 4x + 9 \), compare it to the general form \( y = mx + b \). Here, the coefficient of \( x \) is 4, so the slope \( m \) is 4.
03

Identify the y-intercept

Again, comparing the given equation \( y = 4x + 9 \) to the general form \( y = mx + b \), the constant term is 9, so the y-intercept \( b \) is 9.

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Key Concepts

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

Understanding the Slope
In the world of linear equations, the 'slope' is a key concept. Imagine you're climbing a hill; the steepness of that hill is what we call the slope in mathematics. In the equation y = mx + b, the slope is represented by m. Here's what makes the slope special:
  • It tells us how much the y value changes for a one-unit increase in the x value.
  • A positive slope means the line goes up as you move from left to right.
  • A negative slope means the line goes down as you move from left to right.
For example, in the equation y = 4x + 9, the slope m is 4. This means that for every one unit increase in x, y increases by 4 units. A steep slope like this one indicates a sharp incline.
Understanding the Y-Intercept
The y-intercept is where the line crosses the y-axis. It's the value of y when x is 0. Think of it as the starting point of the line on the y-axis. In the equation y = mx + b, the y-intercept is represented by b. Here's what you need to know about the y-intercept:
  • It is the point where the line touches the y-axis.
  • A positive y-intercept means the line crosses above the origin (0,0).
  • A negative y-intercept means the line crosses below the origin.
For our equation y = 4x + 9, the y-intercept b is 9. This means the line crosses the y-axis at the point (0, 9). So, if you started at the origin and walked up 9 units, you'd be at the y-intercept.
Understanding the General Form of a Linear Equation
Linear equations often follow a standard form which makes them easy to work with and understand. This form is denoted as y = mx + b. Each part of this equation tells us something important about the line:
  • y represents the dependent variable or the output.
  • x represents the independent variable or the input.
  • m is the slope, indicating the steepness of the line.
  • b is the y-intercept, indicating where the line crosses the y-axis.
Knowing this form helps us quickly identify important characteristics. For instance, if you know y = 4x + 9, you can immediately tell:
  • The slope m is 4.
  • The y-intercept b is 9.
This makes graphing and predicting values easier, as you already have crucial information just by looking at the equation.

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