Problem 56 - Algebra 1

Question

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 \(y\) 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 line \(y=\frac{1}{2}x+6\) is \(\frac{1}{2}\) and the y-intercept is 6.

Step 1: Identify The Form of the Equation

Observe that the equation is in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This is known as the slope-intercept form of the line.

Step 2: Find the Slope

The slope of the line is the coefficient of \(x\) in the slope-intercept form. In the equation \(y=\frac{1}{2}x+6\), the slope \(m\) is the coefficient of \(x\), which is \(\frac{1}{2}\). So, the slope of the line is \(\frac{1}{2}\). The slope represents the rate at which the depth of snow increased per hour, which is \(\frac{1}{2}\) inch per hour.

Step 3: Find the Y-intercept

The y-intercept of the line is the constant in the slope-intercept form. In the equation \(y=\frac{1}{2}x+6\), the y-intercept \(b\) is the constant, which is 6. So, the y-intercept is 6. The y-intercept represents the initial depth of snow on the ground before the snowstorm began, which was 6 inches.

Key Concepts

Slope of a Line

Understanding the slope of a line is essential as it describes the steepness and direction of the line. When we talk about a graph on a coordinate plane, the slope tells us how much the 'y' value (the vertical change) goes up or down for every one unit increase in the 'x' value (the horizontal change).

In simpler terms, if you were walking up a hill, the slope would tell you how steep the hill is. The larger the slope, the steeper the hill. Mathematically, slope is typically represented by the letter 'm' and can be calculated when we have the equation of the line in slope-intercept form, which is \(y = mx + b\). The slope-intercept form clearly shows the slope as the coefficient of \(x\).

In our exercise example, the equation \(y=\frac{1}{2}x+6\) is already given in slope-intercept form, making it clear that the slope of the line is \(\frac{1}{2}\), which indicates a gentle rise in y for each unit increase in x. This means that for each hour passed, the depth of the snow increases by half an inch.

Y-intercept

The y-intercept is where a line crosses the y-axis on a graph. This point represents the value of 'y' when 'x' is zero. If you think about starting at the origin and moving straight up or down the y-axis until you hit the line, that's the y-intercept. In the slope-intercept form of a line (\(y = mx + b\)), 'b' represents the y-intercept.

For our snowstorm problem, the equation \(y=\frac{1}{2}x+6\) shows the y-intercept as 6, which conveys that even before any snow fell (at \(x=0\), representing the zero hours mark), there was already a baseline of 6 inches of snow on the ground. The y-intercept, in this case, is a literal starting point before the event (snowfall) occurred.

Linear Equations

Linear equations form straight lines when graphed, and they have consistent growth or decline rates, without curves.A linear equation in two variables (like x and y) can be written in multiple forms, with \(y = mx + b\), the slope-intercept form, being the one that directly shows the slope and y-intercept.

This particular form is very powerful because it can help us easily visualize the graph of the line and make predictions based on the slope and y-intercept. For example, in our exercise, if we wanted to predict the depth of snow after 10 hours, we could plug in \(x=10\) into our equation (\(y=\frac{1}{2}x+6\)) and find out that the depth would be 11 inches.