Sam And Marissa Are Shopping At The Same Store. Sam Bought 2 Model Airplanes And 6 Toy Cars. He Paid $17 Total. Marissa Bought 2 Model Airplanes And Returned 10 Toy Cars. She Paid $9 Total. A System Of Equations That Represents Their Purchases And

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Introduction

In the world of mathematics, systems of equations are a fundamental concept that helps us solve problems involving multiple variables. In this article, we will explore a real-life scenario where two individuals, Sam and Marissa, are shopping at the same store. We will use a system of equations to represent their purchases and solve for the unknown values. By the end of this article, you will understand how to apply systems of equations to solve problems in various fields, including finance, science, and engineering.

The Shopping Scenario

Sam and Marissa are shopping at the same store. Sam bought 2 model airplanes and 6 toy cars. He paid $17 total. Marissa bought 2 model airplanes and returned 10 toy cars. She paid $9 total. Let's represent their purchases using a system of equations.

Variables and Constants

  • Let x be the cost of one model airplane.
  • Let y be the cost of one toy car.
  • Sam's total cost is $17, and Marissa's total cost is $9.

System of Equations

We can represent Sam's and Marissa's purchases using the following system of equations:

  • 2x + 6y = 17 (Sam's total cost)
  • 2x + 10y = 9 (Marissa's total cost)

Solving the System of Equations

To solve this system of equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

Step 1: Multiply the Equations

We will multiply the first equation by 5 and the second equation by 3 to make the coefficients of y equal.

  • 10x + 30y = 85 (First equation multiplied by 5)
  • 6x + 30y = 27 (Second equation multiplied by 3)

Step 2: Subtract the Equations

Now, we will subtract the second equation from the first equation to eliminate the variable y.

  • (10x - 6x) + (30y - 30y) = 85 - 27
  • 4x = 58

Step 3: Solve for x

Now, we will solve for x by dividing both sides of the equation by 4.

  • x = 58/4
  • x = 14.5

Step 4: Substitute x into One of the Original Equations

Now, we will substitute x into one of the original equations to solve for y. We will use the first equation.

  • 2(14.5) + 6y = 17
  • 29 + 6y = 17
  • 6y = -12
  • y = -2

Conclusion

In this article, we used a system of equations to represent the purchases of Sam and Marissa. We solved for the unknown values using the elimination method and found that the cost of one model airplane is $14.50 and the cost of one toy car is -$2. However, since the cost of a toy car cannot be negative, we can conclude that the cost of one toy car is $2.

Real-Life Applications of Systems of Equations

Systems of equations have numerous real-life applications in various fields, including finance, science, and engineering. Here are a few examples:

  • Finance: Systems of equations can be used to model financial transactions, such as investments and loans. For example, a bank may use a system of equations to determine the interest rate on a loan based on the borrower's credit score and loan amount.
  • Science: Systems of equations can be used to model scientific phenomena, such as the motion of objects and the behavior of chemical reactions. For example, a physicist may use a system of equations to model the motion of a projectile based on its initial velocity and angle of launch.
  • Engineering: Systems of equations can be used to design and optimize systems, such as electrical circuits and mechanical systems. For example, an engineer may use a system of equations to design a circuit that meets specific performance requirements.

Tips for Solving Systems of Equations

Here are a few tips for solving systems of equations:

  • Use the elimination method: The elimination method is a powerful tool for solving systems of equations. It involves multiplying the equations by appropriate constants to make the coefficients of one variable equal, and then subtracting the equations to eliminate that variable.
  • Use the substitution method: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.
  • Check your work: It's essential to check your work when solving systems of equations. This involves plugging the solution back into the original equations to ensure that it satisfies both equations.

Conclusion

In this article, we used a system of equations to represent the purchases of Sam and Marissa. We solved for the unknown values using the elimination method and found that the cost of one model airplane is $14.50 and the cost of one toy car is $2. Systems of equations have numerous real-life applications in various fields, including finance, science, and engineering. By following the tips and techniques outlined in this article, you can become proficient in solving systems of equations and apply them to real-world problems.