Problem 5 - Elementary Algebra

Question

Find the equation of each line given the following information. Use the slope- intercept form as the final form of the equation. $$ m=-1, y \text { -intercept }(0,-10) $$

Step-by-Step Solution

Answer

Answer: The equation of the line is y = -x - 10.

Step 1: Recall the slope-intercept form of a linear equation

The slope-intercept form of a linear equation is: $$ y = mx + b $$ where m is the slope and b is the y-intercept of the line.

Step 2: Substitute the values of m and b into the equation

We are given that m = -1 and the y-intercept is (0, -10), so b = -10. Replace m and b in the equation with these values: $$ y = -1x - 10 $$

Step 3: Write the final equation of the line

The equation of the line in slope-intercept form is now: $$ y = -x - 10 $$

Key Concepts

Understanding Linear Equations

Linear equations are equations that produce a straight line when graphed on a coordinate plane. These equations are fundamental in mathematics because they represent a constant rate of change. The most common form of a linear equation is the slope-intercept form known as \( y = mx + b \). In this equation:

• \( y \) is the dependent variable (often representing a real-world quantity, like distance or temperature).
• \( x \) is the independent variable (it typically represents time or another measurable quantity).

Linear equations are the simplest form of equations and are building blocks for more complex mathematical concepts. They allow you to quickly understand how two variables relate to each other by examining the rate of change (slope) and initial value (y-intercept). The graph of a linear equation is always a straight line, hence the name "linear."
Visually, they help in understanding data trends and making predictions.

What is Slope?

The slope of a line is a measure of its steepness and direction. It tells us how much \( y \) changes for every change in \( x \). Slope is noted by the letter \( m \) in the slope-intercept equation \( y = mx + b \). Here's how to understand it practically:

• A positive slope means the line rises as it moves from left to right on the graph.
• A negative slope means the line falls as you travel from left to right.
• A slope of zero means the line is flat, showing no change as \( x \) changes.

In the given exercise, the slope \( m = -1 \), indicating a downward decline as you move across the graph. For each unit increase in \( x \), \( y \) decreases by 1. This negative slope shows a decrease or negative rate of change between the variables in this line.
Understanding slope is key to interpreting the behavior of the linear relationship between two quantities.

The Role of the Y-Intercept

The y-intercept is where the line crosses the \( y \)-axis. This occurs when \( x = 0 \). In the equation \( y = mx + b \), the y-intercept is represented by \( b \). Knowing the y-intercept gives you an starting point of the line on the graph. Here's how it looks in the context of the example:

• You are told the y-intercept is \((0, -10)\), which means the line crosses the \( y \)-axis at -10.
• This point is crucial because it provides a tangible start when plotting the line, showing where \( y \) begins when \( x \) is zero.

In simpler terms, the y-intercept is like the initial value of \( y \) before any increase or decrease caused by the slope begins. It acts as a constant you can rely on regardless of changes in \( x \). In real-world scenarios, this might represent a fixed cost before variable costs are added.
Understanding the y-intercept is essential for making sense of where a line starts on the graph and how to project it further.