Find the slope of the line represented by the equation below.
y = 3 + 2x
The objective is to identify the slope of the line described by the linear equation \(y = 3 + 2x\). A linear equation in two variables \(x\) and \(y\) can often be expressed in the slope-intercept form, which is \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept (the point where the line crosses the y-axis). By comparing our given equation to this standard form, we can directly read off the value of the slope.
To match the standard slope-intercept form \(y = mx + b\), we can use the commutative property of addition to rearrange the terms on the right side of our given equation:
\[y = 3 + 2x = 2x + 3\]
Now, the equation is clearly in the form \(y = mx + b\), where \(m = 2\) and \(b = 3\). The coefficient of \(x\), which is \(2\), represents the slope of the line, indicating that for every unit increase in \(x\), the value of \(y\) increases by 2 units. Thus, the slope of the line is 2.
Understanding the slope of a line is fundamental in various real-life scenarios, such as determining the rate of change. For example, if you are saving money in a piggy bank and you start with \(\$3\) and add \(\$2\) every day, your savings \(y\) after \(x\) days can be modeled by the equation \(y = 3 + 2x\). The slope, which is 2, represents your daily rate of savings. By calculating the slope, you can easily predict how much money you will have at any given time or how quickly your savings are growing.