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Solving For Jody's Summer Earnings Determining Babysitting And Yardwork Hours

Jul 10, 2025 by Admin 78 views

Unraveling Jody's Summer Earnings: A Mathematical Exploration

In this article, we will delve into a classic mathematical problem involving Jody's summer earnings. Jody earns money through two jobs: babysitting and yardwork. She earns $10 per hour babysitting and $15 per hour doing yardwork. This week, she worked a total of 34 hours and earned $410. Our goal is to determine the number of hours Jody spent on each job. To solve this, we will use a system of equations, where '$ x $' represents the number of hours she babysat and '$ y

represents the number of hours she worked in the yard. This problem is a perfect example of how algebra can be used to solve real-world scenarios. By carefully setting up and solving the equations, we can find the exact number of hours Jody dedicated to each job, providing valuable insights into her summer earnings.

To begin, we need to translate the given information into mathematical equations. We know that Jody earns $10 per hour babysitting and $15 per hour doing yardwork. If $x$ represents the number of hours Jody babysits and $y$ represents the number of hours she does yardwork, we can create an equation to represent her total earnings. The earnings from babysitting would be $10x$ (since she earns $10 for each hour), and the earnings from yardwork would be $15y$ (as she earns $15 per hour). The total earnings for the week are $410, so we can write our first equation as:

$10x + 15y = 410$

This equation tells us that the sum of her earnings from babysitting and yardwork equals her total earnings for the week. We also know that Jody worked a total of 34 hours this week. This means that the sum of the hours she spent babysitting ( $x$ ) and the hours she spent doing yardwork ( $y$ ) must equal 34. This gives us our second equation:

$x + y = 34$

Now we have a system of two equations with two variables:

$10x + 15y = 410$
$x + y = 34$

These equations form the foundation for solving our problem. We will use these equations to find the values of $x$ and $y$ , which will tell us how many hours Jody worked at each job. The next step involves choosing a method to solve this system of equations, such as substitution or elimination, which we will explore in the following sections.

Now that we have set up our system of equations, we need to solve for $x$ and $y$ . There are a couple of methods we can use: substitution and elimination. Let's use the substitution method for this problem. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.

Looking at our equations:

$10x + 15y = 410$
$x + y = 34$

The second equation, $x + y = 34$ , looks easier to solve for one of the variables. Let's solve it for $x$ :

$x = 34 - y$

Now we have an expression for $x$ in terms of $y$ . We can substitute this expression into the first equation:

$10(34 - y) + 15y = 410$

This substitution replaces $x$ in the first equation with the equivalent expression in terms of $y$ , giving us an equation with only one variable. Now we can simplify and solve for $y$ :

$340 - 10y + 15y = 410$

Combine like terms:

$5y = 410 - 340$

$5y = 70$

Divide by 5:

$y = 14$

So, Jody worked 14 hours doing yardwork. Now that we have the value of $y$ , we can substitute it back into the equation $x = 34 - y$ to find the value of $x$ :

$x = 34 - 14$

$x = 20$

Therefore, Jody worked 20 hours babysitting. We have now successfully solved the system of equations using the substitution method, determining the number of hours Jody worked at each job.

After solving a system of equations, it's essential to verify our solution to ensure its accuracy. We found that Jody worked 20 hours babysitting ( $x = 20$ ) and 14 hours doing yardwork ( $y = 14$ ). To verify, we can plug these values back into our original equations:

$10x + 15y = 410$
$x + y = 34$

Let's start with the first equation:

$10(20) + 15(14) = 410$

$200 + 210 = 410$

$410 = 410$

The first equation holds true. Now let's check the second equation:

$20 + 14 = 34$

$34 = 34$

The second equation also holds true. This confirms that our solution is correct. Now, let's interpret the solution in the context of the problem. We found that $x = 20$ and $y = 14$ . This means that Jody worked 20 hours babysitting and 14 hours doing yardwork during the week. These values satisfy both the total hours worked (34 hours) and the total earnings ($410). Understanding this solution helps us see how mathematical equations can model and solve real-life scenarios. In this case, we've determined how Jody allocated her time between two jobs to achieve her earnings goal for the week.

While we used the substitution method to solve this problem, it's worth noting that other methods, such as the elimination method, could also be used. The elimination method involves manipulating the equations so that when they are added or subtracted, one of the variables is eliminated. This method can be particularly useful when the coefficients of one variable are multiples of each other.

For example, let's revisit our equations:

$10x + 15y = 410$
$x + y = 34$

To use the elimination method, we could multiply the second equation by -10 to eliminate $x$ :

$-10(x + y) = -10(34)$

$-10x - 10y = -340$

Now, we can add this modified equation to the first equation:

$(10x + 15y) + (-10x - 10y) = 410 + (-340)$

$5y = 70$

$y = 14$

As we found before, $y = 14$ . We can then substitute this value back into one of the original equations to solve for $x$ . Both substitution and elimination are effective methods, and the choice often depends on personal preference or the specific structure of the equations. In addition to these algebraic methods, graphical methods can also be used to solve systems of equations. This involves plotting the equations on a graph and finding the point of intersection, which represents the solution.

In this article, we explored a problem involving Jody's summer earnings, where she worked as a babysitter and did yardwork. We successfully set up and solved a system of equations to determine the number of hours she worked at each job. By translating the given information into mathematical equations, we were able to find that Jody worked 20 hours babysitting and 14 hours doing yardwork. We verified our solution by plugging the values back into the original equations, ensuring its accuracy. This problem illustrates the practical application of algebra in everyday scenarios. Understanding how to set up and solve systems of equations is a valuable skill that can be applied to various real-world situations, from managing finances to planning projects. Moreover, we discussed alternative methods for solving systems of equations, such as the elimination method and graphical methods, highlighting the flexibility and versatility of mathematical problem-solving techniques. Through this exploration, we've gained a deeper understanding of how mathematical concepts can be used to analyze and solve real-world problems effectively.