Dan And Makiko Started Doing Their Homework At The Same Time. It Took Dan Twice As Long To Finish His Homework As It Took Makiko. If Dan Also Took 40 Minutes Longer Than Makiko, Which Of The Following Systems Of Equations Could Be Used To Determine

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Solving Systems of Equations: A Real-World Application

In mathematics, systems of equations are a fundamental concept that helps us solve problems involving multiple variables. In this article, we will explore a real-world scenario where Dan and Makiko are doing their homework, and we need to determine which system of equations can be used to find the time it took each of them to finish their homework.

Dan and Makiko started doing their homework at the same time. It took Dan twice as long to finish his homework as it took Makiko. If Dan also took 40 minutes longer than Makiko, which of the following systems of equations could be used to determine the time it took each of them to finish their homework?

Let's break down the problem and understand what's given:

  • Dan and Makiko start doing their homework at the same time.
  • It took Dan twice as long to finish his homework as it took Makiko.
  • Dan took 40 minutes longer than Makiko.

We can represent the time it took Makiko to finish her homework as x minutes. Since it took Dan twice as long, the time it took Dan to finish his homework is 2x minutes. We are also given that Dan took 40 minutes longer than Makiko, so we can write an equation: 2x = x + 40.

We can represent the problem as a system of equations using the information given. Let's use the variables x and y to represent the time it took Makiko and Dan to finish their homework, respectively.

Since it took Dan twice as long to finish his homework as it took Makiko, we can write the equation:

y = 2x

We are also given that Dan took 40 minutes longer than Makiko, so we can write another equation:

y = x + 40

Now that we have the system of equations, we can solve for x and y. We can start by substituting the expression for y from the first equation into the second equation:

2x = x + 40

Subtracting x from both sides gives us:

x = 40

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:

y = 2x y = 2(40) y = 80

In this article, we explored a real-world scenario where Dan and Makiko are doing their homework, and we needed to determine which system of equations could be used to find the time it took each of them to finish their homework. We represented the problem as a system of equations using the variables x and y to represent the time it took Makiko and Dan to finish their homework, respectively. We then solved the system of equations to find the values of x and y.

The system of equations that could be used to determine the time it took Dan and Makiko to finish their homework is:

y = 2x y = x + 40

x = 40 y = 80

It took Makiko 40 minutes to finish her homework, and it took Dan 80 minutes to finish his homework.

Systems of equations have many real-world applications, including:

  • Finance: Systems of equations can be used to model financial transactions, such as investments and loans.
  • Science: Systems of equations can be used to model scientific phenomena, such as the motion of objects and the behavior of populations.
  • Engineering: Systems of equations can be used to design and optimize systems, such as electrical circuits and mechanical systems.

In conclusion, systems of equations are a fundamental concept in mathematics that can be used to solve problems involving multiple variables. In this article, we explored a real-world scenario where Dan and Makiko are doing their homework, and we needed to determine which system of equations could be used to find the time it took each of them to finish their homework. We represented the problem as a system of equations using the variables x and y to represent the time it took Makiko and Dan to finish their homework, respectively. We then solved the system of equations to find the values of x and `y.
Solving Systems of Equations: A Real-World Application

In our previous article, we explored a real-world scenario where Dan and Makiko are doing their homework, and we needed to determine which system of equations could be used to find the time it took each of them to finish their homework. In this article, we will answer some frequently asked questions about systems of equations.

A system of equations is a set of two or more equations that are related to each other. Each equation in the system is called a linear equation, and the system is called a linear system.

A system of equations can be represented using the variables x and y to represent the time it took Makiko and Dan to finish their homework, respectively. For example:

y = 2x y = x + 40

To solve a system of equations, we need to find the values of the variables x and y that satisfy both equations. We can use various methods to solve a system of equations, including:

  • Substitution method: We can substitute the expression for y from one equation into the other equation.
  • Elimination method: We can add or subtract the equations to eliminate one of the variables.
  • Graphical method: We can graph the equations on a coordinate plane and find the point of intersection.

Systems of equations have many real-world applications, including:

  • Finance: Systems of equations can be used to model financial transactions, such as investments and loans.
  • Science: Systems of equations can be used to model scientific phenomena, such as the motion of objects and the behavior of populations.
  • Engineering: Systems of equations can be used to design and optimize systems, such as electrical circuits and mechanical systems.

The choice of method depends on the specific system of equations and the variables involved. Here are some general guidelines:

  • Substitution method: Use this method when one of the equations is easily solvable for one of the variables.
  • Elimination method: Use this method when the coefficients of the variables are the same in both equations.
  • Graphical method: Use this method when the equations are linear and the point of intersection is easy to find.

Here are some common mistakes to avoid when solving systems of equations:

  • Not checking the solution: Make sure to check the solution by plugging it back into both equations.
  • Not using the correct method: Choose the method that is best suited for the system of equations.
  • Not simplifying the equations: Simplify the equations before solving them to make it easier to find the solution.

In conclusion, systems of equations are a fundamental concept in mathematics that can be used to solve problems involving multiple variables. In this article, we answered some frequently asked questions about systems of equations, including how to represent a system of equations, how to solve a system of equations, and some common mistakes to avoid. We hope this article has been helpful in understanding systems of equations and how to apply them to real-world problems.