Problem 11 - Introductory Algebra for College Students

Question

Find the slope and the \(y\) -intercept of the line with the given equation. $$y=4-x$$

Step-by-Step Solution

Answer

The slope of the line is -1 and the y-intercept is 4.

Step 1: Identify the Form of the Equation

The given equation \(y = 4 - x\) is in the form of a linear equation \(y = mx + c\). Here, we are subtracting \(x\) from \(4\) which can be written as \(y = -x + 4\).

Step 2: Find the Slope

By comparing our equation \(y = -x + 4\) with the standard form, we can identify the coefficient of \(x\) as the slope (m). Therefore, the slope is -1.

Step 3: Find the Y-Intercept

From the equation \(y = -x + 4\), we can identify the y-intercept (c) as the constant term. Therefore, the y-intercept is 4.

Key Concepts

Slope

In the world of linear equations, the slope is a crucial concept. It tells us how steep a line is and what direction it goes. Imagine you're cycling uphill or downhill; the slope is like the angle of the hill.

The slope is represented by the letter \( m \) in the equation \( y = mx + c \). In this mathematical context, slope is defined as the ratio of the "change in \( y \)" over the "change in \( x \)." Often, you'll hear this referred to as "rise over run."

• Positive slope: The line rises as it goes from left to right.
• Negative slope: The line falls as it goes from left to right.
• Zero slope: The line is perfectly flat.
• Undefined slope: The line is perfectly vertical.

In our equation, \( y = -x + 4 \), the slope \( m \) is \(-1\). This means for each step (or unit) you move horizontally, the line goes down 1 unit vertically. This identifies the line as having a downward tilt from left to right.

Y-Intercept

The y-intercept is where the line crosses the y-axis.

This is an important part of understanding where a line sits in relation to a graph. The y-intercept is represented by the letter \( c \) in the equation \( y = mx + c \).In simple terms, it shows the value of \( y \) when \( x \) is zero. Thus, it tells us where the line begins on the y-axis.In our example equation, \( y = -x + 4 \), the y-intercept is \(4\). This means that if you were to graph this equation starting at the y-axis, you would begin right at the point \((0, 4)\). Understanding the y-intercept helps to quickly sketch the graph of the equation and understand how it interacts with the axes.

Linear Functions

Linear functions are all about straight lines.

These functions have equations that can be written in the form \( y = mx + c \). This simple setup makes linear functions popular in math, science, and even everyday problem-solving.A linear function consists of two main components:

• Slope \((m)\) - Defines the steepness and direction of the line.
• Y-Intercept \((c)\) - Defines where the line crosses the y-axis.

These concepts make it easy to predict and understand different linear relationships.
Linear functions are essential because:

• They model relationships with constant rates of change, like speed.
• They make it simple to predict future values.
• They can simplify complex problems into manageable pieces.

When you encounter a linear equation, like \( y = -x + 4 \), knowing these key parts helps you quickly understand the overall behavior and characteristics of the line on the graph.