Problem 39 - Algebra 1: Concepts and Skills

Question

Use the following information. You can work a total of no more than 20 hours per week at your two jobs. Baby-sitting pays 5 dollars per hour, and your job as a cashier pays 6 dollars per hour. You need to earn at least 90 dollars per week to cover your expenses. Give two possible ways you could divide your hours between the two jobs.

Step-by-Step Solution

Answer

Two possible ways to divide the hours could be: 15 hours of babysitting and 5 hours of cashier work; or 14 hours of babysitting and 6 hours of cashier work.

Step 1: Define the Variables

Let \( x \) represent the number of hours you work as a babysitter and \( y \) the number of hours you work as a cashier.

Step 2: Formulate the Constraints

The first constraint is \[ x + y \leq 20 \] since the total working hours should not exceed 20 hours. The second constraint comes from the minimum amount needed per week which is $90. That gives us \[ 5x + 6y \geq 90 \]. The '$5x$' represents income from babysitting and '$6y$' is the income from the cashier job.

Step 3: Find Suitable Solutions

Now, we should find \(x, y\) pairs that satisfy both constraints. For example, if \(x = 15\) and \(y = 5\), both conditions are satisfied since 15 hours of babysitting will earn $75 and 5 hours of cashier work will earn $30 making total $105 for 20 hours of work. Another solution can be \(x = 14\) and \(y = 6\), which will make total $94 for same 20 hours of work.

Key Concepts

Linear Programming

In the context of algebraic problem-solving, linear programming stands out as a powerful method to optimize a given objective function, subject to specific constraints that are represented as linear inequalities. This approach is extensively utilized in various fields such as economics, business, engineering, and more, to find the best outcome in a mathematical model whose requirements are represented by linear relationships.

In the exercise provided, the objective could be to maximize your income from two part-time jobs while adhering to the constraints of your available time and the need to cover weekly expenses. Linear programming would involve plotting the constraints on a graph, finding the feasible region, and then determining the optimal number of hours to work at each job that would yield the highest income without violating the constraints. While the exercise does not explicitly ask for maximizing or minimizing a certain quantity, understanding the basics of linear programming enables students to frame and solve such real-world optimization problems efficiently.

Inequality Constraints

Inequality constraints are fundamental components of linear programming that define the conditions a solution must satisfy. They typically represent limitations or requirements in real-life scenarios, such as time constraints, budget limits, or resource availability.

The exercise presented two inequality constraints: the sum of the hours spent on both jobs should be at most 20 ( \( x + y \leq 20 \) ), and the weekly earnings from both jobs should be at least $90 ( \( 5x + 6y \geq 90 \) ). These inequalities encapsulate the limitations within which any viable solution—the allocation of work hours between both jobs—must fall. To help students better understand these constraints, they can plot them on a coordinate axis system and visualize the solution area as the overlap between the regions that satisfy each inequality, which would then help in identifying multiple solutions or analyzing the trade-offs between different possible scenarios.

Algebraic Problem-Solving

Algebraic problem-solving involves using algebraic methods to find solutions to problems, often translating real-life situations into mathematical models involving variables, equations, and inequalities. This systematic approach begins by defining variables, such as \( x \) for the number of babysitting hours and \( y \) for cashier work hours as given in Step 1 of the solution.

The process continues with the formulation of equations or inequalities that represent the problem's conditions, which in this exercise are represented by Step 2. Solving the problem, or Step 3, can involve various algebraic techniques including graphing, substitution, elimination, or using algebraic formulas. The two provided solutions (\( x = 15 \), \( y = 5 \) and \( x = 14 \), \( y = 6 \) ) are specific instances that satisfy the established constraints, demonstrating how algebra enables us to systematically work through and solve real world issues comprehensively.