Problem 12 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-12,\) find the slope and the \(y\) -intercept of the line with the given equation. $$y=5-x$$

Step-by-Step Solution

Answer

The slope of the line represented by the equation \(y = 5-x\) is \(-1\) and the y-intercept is \(5\).

Step 1: Identify the Slope

The 'slope' of the equation \(y = 5-x\) is the coefficient of \(x\). In this equation, the coefficient of \(x\) is \(-1\), so the slope of this line is \(-1\).

Step 2: Identify the y-intercept

The 'y-intercept' is the constant term in the equation. For the equation \(y = 5-x\), the constant term is \(5\). Therefore, the y-intercept is \(5\).

Key Concepts

Linear equations

Linear equations are an essential topic in algebra and can be thought of as a straight line when graphed on a coordinate plane. These equations can take various forms, but they often appear in the slope-intercept form: \[ y = mx + b \] where:

• \( y \) represents the dependent variable.
• \( x \) is the independent variable.
• \( m \) stands for the slope of the line.
• \( b \) signifies the \( y \)-intercept.

To fully comprehend linear equations, it is essential to understand the role of both the slope and the \( y \)-intercept. Each component provides valuable information about the line's character and position on the graph. By looking at an equation, you can immediately tell the steepness of the line (thanks to the slope) and where it crosses the \( y \)-axis (thanks to the \( y \)-intercept). Together, these elements give you a full picture of your linear equation's graph.

Slope

The slope of a line is a mathematical term that measures how steep the line is. It is defined as the change in \( y \) (the rise) divided by the change in \( x \) (the run) between any two points on the line. This is often expressed as: \[ m = \frac{\Delta y}{\Delta x} \] In the context of a linear equation written in slope-intercept form, the slope is represented by \( m \) in the equation \( y = mx + b \). For the specific equation \( y = 5 - x \), the slope is \(-1\). This means that for every unit increase in \( x \), the value of \( y \) decreases by 1 unit. Think of the slope as the storyteller of the line, revealing its direction and steepness with just a glance. A positive slope means the line ascends from left to right, while a negative slope, like \(-1\), implies the line descends.

Y-intercept

The \( y \)-intercept of a line is the point where the line crosses the \( y \)-axis. In the equation \( y = mx + b \), the \( y \)-intercept is denoted by \( b \). It tells us where the line sits when \( x \) is zero. For the equation \( y = 5 - x \), the \( y \)-intercept is 5. This simply means that when \( x = 0 \), the value of \( y \) is 5. You can quickly locate the \( y \)-intercept on a graph by finding where the line hits the \( y \)-axis. It's the starting or anchor point of your line; no matter what direction the line travels as it moves away from this point, it first touches or begins at the \( y \)-intercept. Understanding the \( y \)-intercept allows one to predict and visualize the line's position and movement along the coordinate plane. Having both the slope and \( y \)-intercept gives you all you need to graph a line or understand its equation.