Problem 11 - Algebra 1

Question

you have \(\$$50 in your savings account at the beginning of the year. Each month you save \)\$ 30\(. Assuming no interest is paid, the equation \)s=30 m+50\( models the amount of money \)s\( in your savings account after \)m\( months. What is the slope of the graph of the line? What is the \)y$ -intercept?

Step-by-Step Solution

Answer

The slope of the graph of the line is 30, and the y-intercept is 50.

Step 1: Identify the Form of the Equation

The equation \(s = 30m + 50\) is in slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Find the Slope

From the equation, we can see that the slope \(m\) of the line is 30. This corresponds to the amount of money saved each month.

Step 3: Find the Y-Intercept

The y-intercept \(b\) of the line is 50, which is the initial amount of money in the savings account.

Key Concepts

Linear Equations

Understanding linear equations is crucial because they describe a relationship where any change in one variable leads to a direct, proportional change in another. In the context of the given exercise, we are dealing with a linear equation that models a real-world situation—saving money over time. The linear equation is presented as \(s=30m+50\), which communicates how the savings account balance, \(s\), changes with each passing month, \(m\). The equation is linear because the relationship between the time and the savings is consistent: each month, the same amount is added to the initial sum.

Linear equations are easily identifiable by their standard form, which is \(y = mx + b\). Here, \(y\) represents the dependent variable while \(x\) represents the independent variable. The constant \(m\) stands for the slope, and \(b\) signifies the y-intercept. These two constants define the unique characteristics of the line. The simplicity of this form allows for quick understanding and calculation of crucial traits of the linear relationship, essential for interpretive and predictive analyses in various fields, including economics, science, and engineering.

Slope of a Graph

The slope of a graph is a measure of how steep a line is and gives insight into the rate of change between the variables represented on the axes. In the example of the savings account, the slope of the graph, symbolized by \(m\) in the equation \(s=30m+50\), is 30. This signifies that for every one-unit increase in \(m\) (which is one month, in this case), the savings \(s\) increase by $30. This consistent increase epitomizes the constant rate of change seen in linear relationships.

Grasping the concept of slope is essential as it appears in various disciplines, not only in mathematics but also in physics (as acceleration or velocity), economics (as marginal cost or benefit), and even in our daily lives (like understanding the inclination of a ramp). To visualize this, imagine plotting the savings for each month on a graph with time on the horizontal axis and the savings amount on the vertical axis. The slope would represent the angle of the line that connects these points—flat for no savings increase or steep as savings accrue faster.

Y-Intercept

The y-intercept is where the line of a graph crosses the y-axis; it's the value of the dependent variable when the independent variable is zero. Referring back to our example, the y-intercept is 50, which corresponds to the initial amount in the savings account before any monthly savings are added. In the equation form \(s=30m+50\), the y-intercept is represented by \(b\).

Understanding y-intercepts is important as they often represent the starting point or initial condition of a scenario being modeled by a linear equation. For instance, in business, the y-intercept could represent the fixed costs before production starts, or in physics, it could signify the initial position of an object before it starts moving. In the graphical representation, where the line crosses the y-axis, that point reflects the scenario's outset, before the influence of other variables comes into play. Recognizing the y-intercept helps in both interpreting data and predicting outcomes for various values of the independent variable.