What is the slope of $y=3x+2$?
The problem asks us to determine the slope of a linear equation given in the form \(y = 3x + 2\). A linear equation in two variables represents a straight line on the Cartesian plane. To identify characteristics like the slope and y-intercept, we compare the given equation to the general slope-intercept form.
The standard slope-intercept form of a linear equation is represented as \(y = mx + b\), where \(m\) represents the slope (the steepness of the line) and \(b\) represents the y-intercept (the point where the line crosses the y-axis). By expressing our equation \(y = 3x + 2\) in this form, we can directly identify the value of the slope \(m\).
Comparing the given equation \(y = 3x + 2\) with the standard form \(y = mx + b\), we can observe that the coefficient of \(x\) is \(3\). This coefficient corresponds to the slope \(m\). Therefore, we have \(m = 3\). Based on the options provided:
The correct option is b.
Consider a taxi service that charges a fixed fee of \(\$2\) just for starting the ride and an additional \(\$3\) for every mile traveled. If \(x\) represents the number of miles, the total cost \(y\) of the taxi ride can be modeled by the linear equation \(y = 3x + 2\). In this scenario, the slope of the line, which is \(3\), represents the cost per mile (the rate of change), while the y-intercept, \(2\), represents the initial flat fee. Understanding this helps in calculating total costs quickly for different trip distances.