Problem 16 - Calculus Single Variable

Question

Write the slope-intercept equation of the line determined by the given data. Slope \(0, y\) -intercept 3

Step-by-Step Solution

Answer

The equation of the line is \( y = 3 \).

Step 1: Recall the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \). In this equation, \( m \) represents the slope of the line, and \( b \) represents the y-intercept.

Step 2: Substitute the Given Slope Value

We substitute the given slope \( m = 0 \) into the slope-intercept form equation. This gives us \( y = 0 \cdot x + b \).

Step 3: Substitute the Given Y-Intercept

Replace the variable \( b \) in the equation \( y = 0 \cdot x + b \) with the given y-intercept value, which is 3. The equation now becomes \( y = 0x + 3 \).

Step 4: Simplify the Equation

Since multiplying by zero results in zero, the term \( 0x \) simplifies to zero. Therefore, the equation simplifies to \( y = 3 \).

Key Concepts

Linear Equations

A linear equation is a statement of equality between two algebraic expressions that involves variables and constants. These equations can be represented in various forms, and one common form is the slope-intercept form, known for its simplicity and clarity. The general linear equation is typically expressed as

• ax + by = c
• y = mx + b

but the slope-intercept form is particularly useful for understanding the geometric properties of the line on a Cartesian plane. Linear equations describe a straight line, which is essential in various fields such as physics, economics, and more. They simplify problem-solving by allowing predictions and interpretations of data. The parameters within a linear equation give us valuable information, like the direction in which the line tilts (slope) and where the line crosses the y-axis (y-intercept). Understanding linear equations helps in analyzing relationships and trends between different quantities.

Slope of a Line

The slope of a line indicates its steepness or incline and is usually represented by the letter "m." It is calculated as the ratio of the change in the y-coordinate to the change in the x-coordinate between any two points on the line.

• Slope formula: rac{y_2 - y_1}{x_2 - x_1}
• Horizontal lines have a slope of zero, as they do not rise or fall.
• Vertical lines have an undefined slope because their x-coordinates do not change.

A positive slope means the line ascends as it moves from left to right, while a negative slope denotes a descending line. Recognizing and interpreting the slope helps in understanding how variables are related. For example, a slope of zero, like in our exercise, means we have a horizontal line which models a situation where the dependent variable remains constant regardless of changes in the independent variable. This is a crucial part of algebra, where the slope helps in determining linear trends and making sense of practical situations.

Y-Intercept

The y-intercept is a vital part of understanding the graph of a linear equation. It is the point where the line crosses the y-axis, indicating the value of y when x is zero.

• The y-intercept is represented by the letter "b" in the slope-intercept equation, y = mx + b.
• It helps to quickly identify the starting point of a line on the graph.
• In the example, the y-intercept is 3, meaning the line crosses the y-axis at (0, 3).

Understanding the y-intercept allows us to determine the initial value or starting condition of a situation modeled by the line. It is particularly useful in scenarios requiring interpretation of data starting from a baseline, such as in economics or natural sciences. The y-intercept provides a point of reference which, when combined with the slope, gives a complete picture of the linear relationship in question.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It serves as a foundation for solving equations and finding unknown values. It includes concepts such as variables, coefficients, expressions, and equations. Algebra allows us to describe and solve real-world problems using mathematical models. Key concepts in algebra include:

• Understanding variables as placeholders for numbers.
• Using expressions to represent real-world quantities.
• Solving equations to find unknown values.
• Graphing equations to visualize relationships between variables.

By using algebra, we transform complex problems into easily manageable formats. The use of the slope-intercept form is an example of applying algebra to understand the nature of lines and predict points on a graph. Algebra's principles aid in cultivating logical thinking and can be applied in various fields like engineering, economics, science, and more. It is an essential tool for interpreting and solving both simple and complex problems across many disciplines.