Determining the System of Equations

Oct 13, 2025 by miodeya.com 27 views

Introduction

In this article, we will explore how to model a real-world situation using systems of equations. We will use the example of Lizzy, who has 30 coins that total $\$4.80$ . All of her coins are dimes, $D$ , and quarters, $Q$ . We will determine which system of equations models this situation.

Understanding the Problem

Lizzy has 30 coins that total $\$4.80$ . The coins are a combination of dimes and quarters. We can represent the number of dimes as $D$ and the number of quarters as $Q$ . Since the total value of the coins is $\$4.80$ , we can set up an equation to represent this situation.

Setting Up the Equations

Let's start by setting up an equation to represent the total number of coins. We know that Lizzy has 30 coins, so we can write an equation:

D + Q = 30

This equation represents the total number of coins.

Next, we need to set up an equation to represent the total value of the coins. We know that the total value of the coins is $\$4.80$ , and we can represent the value of each coin as follows:

A dime is worth $\$0.10$ , so the total value of the dimes is $0.10D$ .
A quarter is worth $\$0.25$ , so the total value of the quarters is $0.25Q$ .

We can set up an equation to represent the total value of the coins:

0.10D + 0.25Q = 4.80

This equation represents the total value of the coins.

Determining the System of Equations

Now that we have set up the equations, we can determine which system of equations models this situation. We have two equations:

D + Q = 30

0.10D + 0.25Q = 4.80

We can represent these equations as a system of equations:

{ \begin{array}{l} D + Q = 30 \\ 0.10D + 0.25Q = 4.80 \end{array} \}

This system of equations models the situation of Lizzy having 30 coins that total $\$4.80$ .

Conclusion

In this article, we have explored how to model a real-world situation using systems of equations. We used the example of Lizzy, who has 30 coins that total $\$4.80$ . We set up two equations to represent the total number of coins and the total value of the coins, and we determined which system of equations models this situation. The system of equations is:

{ \begin{array}{l} D + Q = 30 \\ 0.10D + 0.25Q = 4.80 \end{array} \}

This system of equations can be used to solve for the number of dimes and quarters that Lizzy has.

Discussion

This problem can be solved using substitution or elimination methods. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable.

Example Solution

Let's use the substitution method to solve this system of equations. We can solve the first equation for $D$ :

D = 30 - Q

We can substitute this expression into the second equation:

0.10(30 - Q) + 0.25Q = 4.80

Simplifying this equation, we get:

3 - 0.10Q + 0.25Q = 4.80

Combine like terms:

0.15Q = 1.80

Divide both sides by 0.15:

Q = 12

Now that we have found the value of $Q$ , we can substitute this value back into the first equation to find the value of $D$ :

D + 12 = 30

Subtract 12 from both sides:

D = 18

Therefore, Lizzy has 18 dimes and 12 quarters.

Conclusion

{ \begin{array}{l} D + Q = 30 \\ 0.10D + 0.25Q = 4.80 \end{array} \}

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are related to each other. In this case, we have two equations that represent the total number of coins and the total value of the coins.

Q: How do I know which system of equations to use?

A: To determine which system of equations to use, you need to identify the variables and the relationships between them. In this case, we have two variables: the number of dimes (D) and the number of quarters (Q). We also have two equations that represent the total number of coins and the total value of the coins.

Q: What is the difference between a system of equations and a single equation?

A: A single equation is a statement that expresses a relationship between two or more variables. A system of equations, on the other hand, is a set of two or more equations that are related to each other. In this case, we have two equations that represent the total number of coins and the total value of the coins.

Q: How do I solve a system of equations?

A: There are several methods to solve a system of equations, including substitution and elimination. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable.

Q: What is the substitution method?

A: The substitution method is a method of solving a system of equations by solving one equation for one variable and then substituting that expression into the other equation. In this case, we solved the first equation for D and then substituted that expression into the second equation.

Q: What is the elimination method?

A: The elimination method is a method of solving a system of equations by adding or subtracting the equations to eliminate one variable. In this case, we could have added the two equations together to eliminate the variable Q.

Q: How do I know which method to use?

A: The choice of method depends on the specific problem and the variables involved. In this case, we used the substitution method because it was easier to solve the first equation for D and then substitute that expression into the second equation.

Q: What if I have more than two equations?

A: If you have more than two equations, you can use the same methods to solve the system of equations. However, you may need to use more advanced techniques, such as matrix operations or graphing.

Q: Can I use a calculator to solve a system of equations?

A: Yes, you can use a calculator to solve a system of equations. Many calculators have built-in functions for solving systems of equations, including the substitution and elimination methods.

Q: What if I get stuck or make a mistake?

A: If you get stuck or make a mistake, don't worry! You can always go back and try again. You can also ask for help from a teacher, tutor, or classmate.

Conclusion

In this article, we have answered some frequently asked questions about systems of equations. We have discussed the difference between a system of equations and a single equation, and we have explained the substitution and elimination methods for solving a system of equations. We have also provided some tips for choosing the right method and for using a calculator to solve a system of equations.