Modeling Jody's Earnings With Equations
Introduction
In the realm of mathematical problem-solving, real-world scenarios often present themselves as intricate puzzles waiting to be unraveled. Consider the situation of Jody, a diligent individual who juggles two summer jobs babysitting and yardwork to make the most of her time and earn a decent income. This article delves into a mathematical exploration of Jody's work situation, where she earns $10 per hour babysitting and $15 per hour doing yardwork. This week, she dedicated a total of 34 hours to these jobs and earned $410. Our objective is to dissect this scenario using algebraic equations, with x representing the hours spent babysitting and y representing the hours spent on yardwork. We will embark on a step-by-step journey to formulate and solve these equations, shedding light on the number of hours Jody devoted to each job. This problem not only sharpens our algebraic skills but also illustrates the practical application of mathematics in everyday financial planning and time management.
The challenge before us is to determine precisely how many hours Jody dedicated to babysitting and how many to yardwork. This requires translating the given information into a system of equations, a cornerstone of algebraic problem-solving. The total hours worked and the total earnings provide us with two critical pieces of information, which we can then convert into mathematical expressions. By carefully analyzing these expressions, we will be able to construct a system of equations that accurately models Jody's work situation. The subsequent steps will involve employing algebraic techniques to solve this system, revealing the specific number of hours Jody spent on each job. This exploration is not merely an academic exercise; it mirrors the kind of financial decisions individuals make regularly, highlighting the relevance of mathematical reasoning in personal and professional contexts. Through this detailed analysis, we aim to provide a comprehensive understanding of how mathematical tools can be applied to solve real-world problems, emphasizing the importance of clear problem formulation, strategic equation solving, and accurate interpretation of results.
Setting Up the Equations
To effectively decipher Jody's work schedule, the crucial first step involves translating the given information into a set of algebraic equations. The core concept here is to represent the unknown quantities the hours Jody spent babysitting and doing yardwork using variables. As stated, let x denote the number of hours Jody worked as a babysitter, earning $10 per hour, and let y represent the hours she spent doing yardwork, where she earns $15 per hour. We have two key pieces of information to work with the total number of hours Jody worked and her total earnings for the week. The total hours worked is the sum of hours spent babysitting (x) and hours spent on yardwork (y), which amounts to 34 hours. This can be directly translated into our first equation: x + y = 34. This equation is a linear equation in two variables, a fundamental component of our system of equations.
Our second piece of information pertains to Jody's total earnings. She earns $10 for each hour of babysitting, so her total earnings from babysitting are 10x. Similarly, she earns $15 for each hour of yardwork, contributing 15y to her total earnings. The sum of these two amounts represents her total earnings for the week, which is given as $410. Therefore, we can formulate the second equation as 10x + 15y = 410. This equation, like the first, is also a linear equation in two variables, but it incorporates the hourly rates to reflect Jody's earnings from each job. Together, these two equations form a system of linear equations:
- x + y = 34
- 10x + 15y = 410
This system of equations is the mathematical representation of Jody's work situation. Solving this system will provide us with the values of x and y, revealing the exact number of hours Jody worked at each job. The next phase involves selecting an appropriate method to solve this system, which will be discussed in the subsequent section. The ability to translate real-world scenarios into mathematical equations is a crucial skill in problem-solving, and this example aptly demonstrates this process.
Solving the System of Equations
With the system of equations firmly established, the next pivotal step is to solve it to determine the values of x and y, representing the hours Jody spent babysitting and doing yardwork, respectively. There are several methods available for solving systems of linear equations, including substitution, elimination, and graphical methods. For this particular problem, we will employ the method of substitution, which is particularly effective when one of the equations can be easily rearranged to express one variable in terms of the other. Looking at our system of equations:
- x + y = 34
- 10x + 15y = 410
Equation 1 appears simpler and can be readily rearranged to isolate one of the variables. Let's solve Equation 1 for x: x = 34 - y. This expression now allows us to substitute for x in Equation 2, effectively reducing the system to a single equation with a single variable. Substituting (34 - y) for x in Equation 2 yields:
10*(34 - y) + 15y = 410
Now, we have an equation with only one variable, y, which we can solve using standard algebraic techniques. First, distribute the 10 across the terms inside the parentheses:
340 - 10y + 15y = 410
Next, combine like terms involving y:
340 + 5y = 410
Subtract 340 from both sides of the equation to isolate the term with y:
5y = 410 - 340
5y = 70
Finally, divide both sides by 5 to solve for y:
y = 14
We have now determined that Jody spent 14 hours doing yardwork. To find the number of hours she spent babysitting, we substitute y = 14 back into the expression we derived earlier, x = 34 - y:
x = 34 - 14
x = 20
Therefore, Jody spent 20 hours babysitting. We have successfully solved the system of equations, revealing the number of hours Jody dedicated to each of her summer jobs. The next step involves verifying these solutions to ensure their accuracy and to provide a conclusive answer to the problem.
Verifying the Solution
After obtaining potential solutions to a system of equations, it is crucial to verify their accuracy. This verification process ensures that the solutions satisfy all the original conditions and equations provided in the problem. In our scenario, we found that Jody spent 20 hours babysitting (x = 20) and 14 hours doing yardwork (y = 14). To verify these solutions, we will substitute these values back into the original equations:
- x + y = 34
- 10x + 15y = 410
First, let's substitute the values into Equation 1:
20 + 14 = 34
34 = 34
The equation holds true, confirming that the total hours worked add up correctly. Now, let's substitute the values into Equation 2:
10*(20) + 15*(14) = 410
200 + 210 = 410
410 = 410
This equation also holds true, indicating that the total earnings calculated from the hours worked at each job match the given total earnings. Since both equations are satisfied by our solutions x = 20 and y = 14, we can confidently conclude that these values are correct. The verification process is a critical step in mathematical problem-solving, as it confirms the accuracy of the solutions and ensures that they align with the problem's constraints. This step not only validates the mathematical calculations but also provides assurance that the answers are meaningful within the context of the real-world situation.
Conclusion
In summary, this article has meticulously dissected a real-world problem involving Jody's summer job earnings, employing algebraic techniques to arrive at a definitive solution. We began by translating the problem's narrative into a system of linear equations, representing the number of hours Jody worked babysitting (x) and doing yardwork (y) as variables. The two key pieces of information the total hours worked (34) and the total earnings ($410) were used to construct the equations:
- x + y = 34
- 10x + 15y = 410
We then employed the substitution method to solve this system. By solving the first equation for x (x = 34 - y) and substituting this expression into the second equation, we reduced the problem to a single equation with one variable. Solving for y revealed that Jody worked 14 hours doing yardwork. Substituting this value back into the equation x = 34 - y gave us x = 20, indicating that Jody worked 20 hours babysitting.
To ensure the accuracy of our solutions, we performed a verification step, substituting x = 20 and y = 14 back into the original equations. Both equations held true, confirming the correctness of our solutions. Therefore, we can confidently conclude that Jody spent 20 hours babysitting and 14 hours doing yardwork. This problem serves as a practical illustration of how algebraic equations can be used to model and solve real-world situations involving time management and earnings. The ability to translate word problems into mathematical expressions and solve them is a valuable skill, applicable in various fields and everyday scenarios. Through this detailed analysis, we have not only solved a specific problem but also highlighted the broader importance of mathematical reasoning in problem-solving and decision-making.
Select the correct answer
Question: During the summer, Jody earns $10 per hour babysitting and $15 per hour doing yardwork. This week she worked 34 hours and earned $410. If x represents the number of hours she babysat and y represents the number of hours she did yardwork, which system of equations models this situation?
