Problem 25 - Contemporary Precalculus

Question

Find the slope and y-intercept of the line whose equation is given. $$2 x-y+5=0$$

Step-by-Step Solution

Answer

Answer: The slope of the line is \(2\), and the y-intercept is \(-5\).

Step 1: Rewrite the given equation in slope-intercept form

First, we need to isolate y in the given equation. Start by moving the \(2x\) term and constant from the left side to the right side of the equation: $$-y = -2x + 5$$ Next, multiply both sides of the equation by \(-1\) to make the \(y\) term positive: $$y = 2x - 5$$ Now, the equation is in slope-intercept form.

Step 2: Identify the slope and y-intercept

In the slope-intercept form equation, the coefficient of \(x\) is the slope and the constant term is the y-intercept. In this case: $$y = 2x - 5$$ It's clear that the slope \(m = 2\) and the y-intercept \(b = -5\). Thus, the slope of the line is \(2\), and the y-intercept is \(-5\).

Key Concepts

Linear Equations

Linear equations are a central concept in algebra. These are equations in which the highest power of the variable is one. Typically, linear equations are written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. The importance of linear equations lies in their simplicity and ubiquity in modeling real-world situations. Linear equations often describe relationships between two variables, such as distance and time or cost and quantity.

When dealing with linear equations, converting them into different forms can make the information more accessible or solve problems more efficiently. One of the most useful forms is the slope-intercept form, which makes it easy to identify and visualize important characteristics of the line it represents. By understanding the structure of a linear equation, you can easily convert it into the slope-intercept form to find the slope and y-intercept.

Slope

The slope of a line is a numerical measure of its steepness and direction. In the context of the slope-intercept form of a linear equation, which is \(y = mx + b\), the slope is represented by the coefficient \(m\).

Understanding the slope helps you determine how the line behaves:

• Positive slope: The line rises as it moves from left to right.
• Negative slope: The line falls as it moves from left to right.
• Zero slope: The line is horizontal, indicating no change in \(y\) as \(x\) varies.
• Undefined slope: This applies to vertical lines where \(x\) remains constant.

To find the slope from a given linear equation, transform the equation into the slope-intercept form \(y = mx + b\). In this form, \(m\) provides an immediate understanding of how much \(y\) changes with \(x\). For example, a slope of \(2\) means that for every increase of \(1\) in \(x\), \(y\) increases by \(2\). This straightforward interpretation is why mastering the slope is crucial for analyzing linear relationships.

Y-intercept

The y-intercept is a fundamental part of understanding the graph of a linear equation. It is the point where the line crosses the y-axis, represented by \(b\) in the slope-intercept form \(y = mx + b\).

To comprehend it better:

• The y-intercept shows the value of \(y\) when \(x\) is zero.
• It provides a starting point for graphing the line.
• A positive y-intercept means the line crosses above the origin; a negative y-intercept, below the origin.

To identify the y-intercept from a linear equation, rearrange the equation into the slope-intercept form if necessary. The constant term \(b\) will give you this intersection point.

For instance, in the equation \(y = 2x - 5\), the y-intercept is \(-5\). This means the line crosses the y-axis at \(-5\), forming a foundational understanding of the line's position in the coordinate plane.