Problem 38 - Understanding Elementary Algebra with Geometry

Question

Determine the slope of the line from its equation. $$y=2 x-11$$

Step-by-Step Solution

Answer

The slope is 2.

Step 1: Identify the equation form

Recognize that the equation of the line is given in the slope-intercept form, which is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Locate the slope in the equation

Compare the given equation \(y = 2x - 11\) with the slope-intercept form \(y = mx + b\). Identify the coefficient of \(x\) as the slope.

Step 3: Extract the slope

From the comparison, the coefficient of \(x\) is found to be 2. Therefore, the slope \(m\) is 2.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most commonly used forms. It is represented as \(y = mx + b\). This form is beneficial in that it directly gives both the slope and the y-intercept of the line. The slope (\frac{∆y}{∆x}\text{rise over run}), noted as \(m\), shows how steep the line is. The y-intercept (\text{b}), is the point where the line crosses the y-axis (\text{value of y when x = 0}). Recognizing this form in equations makes determining these two key features very straightforward.

Linear Equations

Linear equations are equations of the first degree. This means the highest exponent the variable can have is 1. They can be written in several forms, such as slope-intercept form \(y = mx + b\), standard form \(Ax + By = C\), and point-slope form \(y - y_1 = m(x - x_1)\). Regardless of the form, all linear equations plot straight lines when graphed. Understanding how to transform any form into the slope-intercept form can make analysis easier.

Identifying Slope

Identifying the slope of a line is crucial for many mathematical applications. In the slope-intercept form \(y = mx + b\), the slope is the coefficient of \(x\). It indicates the line's direction and steepness. For instance, in the equation \(y = 2x - 11\), by comparing it to \(y = mx + b\), it is clear that the slope \(m = 2\). A positive slope means the line ascends from left to right, while a negative slope means the line descends.