Problem 58 - Contemporary Precalculus

Question

Find an equation for the line satisfying the given conditions. \(y\) -intercept -7 and slope 1.

Step-by-Step Solution

Answer

Answer: The equation of the line is $$y = x - 7$$.

Step 1: Identify the slope and y-intercept

We are given the slope \(m = 1\) and the y-intercept \(b = -7\).

Step 2: Use the slope-intercept form

The slope-intercept form of a linear equation is \(y = mx + b\). Plug in the given values of the slope (m) and y-intercept (b) into the equation: \(y = 1x -7\).

Step 3: Simplify the equation if necessary

As our equation is already in its simplest form, no further simplification is necessary: \(y = 1x -7\).

Step 4: Write down the final answer

The equation of the line with a slope of 1 and a y-intercept of -7 is: $$y = x - 7$$.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is fundamental when dealing with linear equations. It's represented as the equation of a line, which can be written as \( y = mx + b \). In this equation, \( m \) corresponds to the slope of the line, that is, how steep the line is. A higher value of \( m \) indicates a steeper incline, while a negative slope suggests a decline. The slope is also the rate at which \( y \) changes for a unit change in \( x \).

The second important component, \( b \), represents the y-intercept, which tells us where exactly our line crosses the y-axis. This is the value of \( y \) when \( x \) is zero. This simple structure \( y = mx + b \) allows us to quickly sketch a graph of the line when given these two crucial pieces of information or to determine the slope and position based on a graph.

Linear Equations

A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. Linear equations can come in several forms, with the slope-intercept form being one of the most commonly used because of its simplicity. Linear equations describe straight lines in a two-dimensional space.

The applications of linear equations are vast, ranging from predicting profits in a business model to calculating the distance covered by an object moving at a constant speed. The beauty of linear equations lies in their direct proportionality, meaning they have a constant rate of change, which is visually represented by a straight line on the graph.

Y-Intercept

The y-intercept of a line is a fundamental concept in graphing linear equations. It's the point at which the line crosses the y-axis of a coordinate plane. In the equation \( y = mx + b \), the y-intercept is represented by \( b \).

In practical terms, the y-intercept gives us an initial value or starting point of the function or real-world scenario being modeled. For instance, in a business setting, the y-intercept could represent the starting amount of profit or loss before any units are sold (when \( x = 0 \)). Understanding the y-intercept helps in both graphing lines and interpreting the graphs in context.