Problem 30 - Algebra 1

Question

Write a linear equation to model the situation. Use unit analysis to check your model. You borrow \(\$ 40\) from your sister. To repay the loan, you pay her \(\$ 5\) a week.

Step-by-Step Solution

Answer

The linear equation that models the loan situation is \(y = -5x + 40\)

Step 1: Identify the y-intercept and slope

The initial borrowed amount is \$40. This means when no week has passed (week 0), the borrowed amount is \$40. Hence, y-intercept is \$40. The loan is being repaid at \$5 per week. This value is negative because it reduces the outstanding loan. Hence, the slope is -\$5.

Step 2: Formulate the linear equation

A linear equation can be formulated in the form \(y = mx + c\) where \(m\) is the slope and \(c\) is the y-intercept. Here \(y\) is the remaining loan amount after \(x\) weeks, \(m = -5\) and \(c = 40\). Plugging these into the equation gives \(y = -5x + 40\).

Step 3: Unit analysis

Check if the units on both sides of the equation are consistent. In our case, \(y\) is the remaining loan amount after \(x\) weeks. On the right side, \(5x\) gives a result in dollars (as it represents the money paid) and 40 is also in dollars (as it represents the initial borrowed amount). Hence, the equation is consistent in terms of units.

Key Concepts

Slope

In the context of a linear equation, the slope is a crucial concept that determines the steepness or incline of the line on a graph. In simpler terms, it tells you how much one variable changes when the other variable changes. Slope can be thought of as the "rate of change".

To find the slope, you calculate the difference in the y-values divided by the difference in the x-values between two points on the line. It is usually represented by the letter "m" in the equation of a line, which typically looks like this: \(y = mx + c\).

In our exercise, the slope is \\(-5\\), meaning each week (x) the amount you owe (y) decreases by \\(5\\). This negative sign indicates that the debt decreases as each week passes.

Y-intercept

The y-intercept is another fundamental part of a linear equation. It represents the point where the line crosses the y-axis on a graph. This point is when the value of x is zero.

In practical terms, the y-intercept is the initial value or starting point of your function. It's the amount of something at the beginning, before any changes begin to occur.

For our exercise, the y-intercept is \\(40\\), because this is the amount of money initially borrowed before any repayments start. When you set week 0 (x=0), you can see from the equation, \(y = -5x + 40\), that y remains 40.

Units Analysis

Units analysis is a way to ensure that your equation makes sense in terms of the units being used. It provides a check to make sure that all parts of the equation match up appropriately in terms of their measurement units, such as dollars, meters, seconds, etc.

This concept involves checking each term in your equation to see if they make sense when calculated together. For instance, if the units on one side of the equation don't match the units on the other side, something is probably wrong.

In our example, units analysis confirms the equation \(y = -5x + 40\) fits properly. Both \\(-5x\\) (the repayment in dollars) and \\(40\\) (the initial borrowed money in dollars) deal with money, so they share consistent units of dollars.

Algebraic Modeling

Algebraic modeling involves creating mathematical representations of real-world situations using equations. By abstracting a problem into an equation, you can solve it to predict how a system behaves under different conditions.

In the given exercise, algebraic modeling is used to represent the debt repayment scenario by forming a linear equation. This involves identifying key components—like the slope and y-intercept—to fit the scenario into an \(y = mx + c\) format.

Modeling real-life issues algebraically, as in this example, not only provides a clearer understanding of numeric relationships but also assists in planning and decision-making. Here, it helps visualize how much money remains to be paid each week, allowing for informed budgeting and financial decisions.