Problem 24 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ y=2 x+9 $$

Step-by-Step Solution

Answer

Answer: The slope is \(2\) and the y-intercept is \(9\).

Step 1: Identify the given equation

The given equation is: $$ y = 2x + 9 $$ This equation is already in the form of a linear function.

Step 2: Identify the slope

The slope \(m\) is the coefficient of the \(x\) term. In the given equation, the slope is: $$ m = 2 $$

Step 3: Identify the y-intercept

The y-intercept \(b\) is given by the constant term in the equation. In the given equation, the y-intercept is: $$ b = 9 $$

Step 4: State the slope and y-intercept

The slope of the line is \(2\), and the y-intercept is \(9\). Therefore, the given linear function can be expressed as: $$ y = 2x + 9 $$

Key Concepts

Slope

The slope is a fundamental aspect of a linear function. It measures how steep a line is. More technically, the slope indicates the rate at which the value of the dependent variable (usually y) changes in response to changes in the independent variable (usually x). The slope is usually represented by the letter \( m \) in equations.
For example, in the equation \( y = 2x + 9 \), the slope \( m \) is 2. This means that for every unit increase in \( x \), the value of \( y \) increases by 2 units.

• If the slope is positive, the line rises from left to right.
• If the slope is negative, the line falls from left to right.
• A larger slope means a steeper line.
• If the slope is zero, the line is horizontal, indicating no change in \( y \).

Understanding the concept of slope is crucial for analyzing and graphing linear equations.

Y-Intercept

The y-intercept is the point at which the line crosses the y-axis. It is an essential characteristic for graphing a linear function because it provides a starting point. Represented by \( b \) in the equation of a line, it indicates where the line will hit the y-axis when \( x = 0 \).
In the equation \( y = 2x + 9 \), the y-intercept \( b \) is 9. This means that when \( x \) equals 0, \( y \) is 9. Therefore, the line crosses the y-axis at the point (0, 9).

• The y-intercept allows us to easily graph a line when combined with the slope.
• It provides a reference for determining the position of a line on a graph.
• The y-intercept is often used to compare different lines easily.

Using the y-intercept simplifies sketching and understanding linear graphs.

Linear Function

A linear function is a mathematical expression that creates a straight line when graphed. It has the general form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Linear functions are used to model relationships with constant rates of change.
In the equation \( y = 2x + 9 \), this is a linear function because it fits the \( y = mx + b \) form, with its slope \( m = 2 \) and y-intercept \( b = 9 \).

• Linear functions have no exponents higher than one, making them straight lines.
• They can represent real-world situations like calculating costs, speed, or growth over time.
• Linear functions are foundational concepts in algebra, allowing for analysis and interpolation of trends.

When you understand linear functions, you can predict values and comprehend relationships within the data.

Coefficient

In the context of linear equations, the coefficient is the number that multiplies a variable. Coefficients tell us about the relationship between variables within the equation, especially in the context of slope and intercept calculations.
For example, in \( y = 2x + 9 \), the coefficient is 2. This coefficient is crucial because it indicates the responsiveness of \( y \) to changes in \( x \), often presented as rise over run in a graphical representation of a line.

• The coefficient of \( x \) is always the slope in a linear equation.
• It dictates the steepness and direction of the line on a graph.
• Understanding coefficients allows for manipulation and transformation of equations for various applications.

Learning about coefficients gives insights into how variables influence each other in mathematical equations.