Problem 56 - Algebra 1: Concepts and Skills

Question

Use the following information. Snow fell for 9 hours at a rate of \(\frac{1}{2}\) inch per hour. Before the snowstorm began, there were already 6 inches of snow on the ground. The equation \(y=\frac{1}{2} x+6\) models the depth y (in inches) of snow on the ground after x hours. What is the slope of\(y=\frac{1}{2} x+6 ?\)What is the y-intercept?

Step-by-Step Solution

Answer

The slope of the equation \(y=\frac{1}{2} x+6\) is \(\frac{1}{2}\) and the y-intercept is 6.

Step 1: Identify the Slope

The slope of a linear equation is the coefficient of x. In the equation \(y=\frac{1}{2} x+6\), the coefficient of x (which is the slope) is \(\frac{1}{2}\). Thus, the slope is \(\frac{1}{2}\).

Step 2: Identify the Y-Intercept

The y-intercept of a linear equation is the constant term. In the equation \(y=\frac{1}{2} x+6\), the constant term (which is the y-intercept) is 6. Thus, the y-intercept is 6.

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial when dealing with linear equations. The slope is essentially the measure of how steep a line is. It's often referred to as the line's 'rise over run' and is calculated by determining the change in the 'y' values over the change in the 'x' values between two distinct points on the line.

For example, in the equation of a line given by the formula:
\(y = mx + b\), 'm' represents the slope of the line. This number tells us how much the value of 'y' increases or decreases as 'x' increases by 1. In the context of the exercise, where we have the equation \(y = \frac{1}{2}x + 6\), the slope is \(\frac{1}{2}\). This means for every hour that passes (increase in 'x'), the depth of the snow increases by half an inch.

Y-Intercept

The y-intercept is another fundamental component of linear equations. It is the value of 'y' when the line crosses the y-axis. In other words, it's the point at which 'x' is zero.

In the equation \(y = mx + b\), 'b' represents the y-intercept. Thus, if we have a look at the equation from the exercise, \(y = \frac{1}{2}x + 6\), the y-intercept is 6. This indicates the starting point of the situation described by the equation before any changes are made to 'x'. In this scenario, it tells us that before the snow started falling, there were already 6 inches of snow on the ground.

Algebraic Modeling

Algebraic modeling is a method of using algebra to represent real-life situations. By defining variables and creating equations, we can explore and solve real-world problems systematically.

In the given exercise, we modeled the depth of snow over time with the equation \(y = \frac{1}{2}x + 6\). Here, 'y' represents the total depth of snow on the ground after 'x' hours of snowfall at a constant rate. The beauty of algebraic modeling lies in its ability to make predictions. For instance, if we want to know the expected snow depth after 3 more hours, we can simply plug '3' into our equation for 'x' and solve for 'y'. As real situations often change, the ability to modify these models is crucial for them to remain accurate and useful tools.