Martin Has A Combination Of 33 Quarters And Dimes Worth A Total Of $$ 6 6 6 $. Which System Of Linear Equations Can Be Used To Find The Number Of Quarters, $q$, And The Number Of Dimes, $d$, Martin Has?A.

Mar 11, 2025 by ADMIN 209 views

**Solving a System of Linear Equations: Finding the Number of Quarters and Dimes**

Introduction

In this article, we will explore a system of linear equations that can be used to find the number of quarters and dimes Martin has. The problem states that Martin has a combination of 33 quarters and dimes worth a total of $6. We will use algebraic methods to set up a system of linear equations and solve for the number of quarters and dimes.

Understanding the Problem

Let's break down the problem and understand what we are trying to find. We know that Martin has a combination of quarters and dimes worth a total of $6. We also know that there are 33 coins in total. Our goal is to find the number of quarters, denoted by q, and the number of dimes, denoted by d.

Setting Up the System of Linear Equations

To set up the system of linear equations, we need to consider the value of each coin. A quarter is worth $0.25, and a dime is worth $0.10. We can set up two equations based on the total value and the total number of coins.

Equation 1: Total Value

The total value of the coins is $6, which can be represented by the equation:

0.25q + 0.10d = 6

Equation 2: Total Number of Coins

The total number of coins is 33, which can be represented by the equation:

q + d = 33

Solving the System of Linear Equations

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

Step 1: Multiply Equation 2 by 0.25

To eliminate the variable q, we can multiply Equation 2 by 0.25:

0.25(q + d) = 0.25(33) 0.25q + 0.25d = 8.25

Step 2: Subtract Equation 1 from the Result

Now, we can subtract Equation 1 from the result:

(0.25q + 0.25d) - (0.25q + 0.10d) = 8.25 - 6 0.15d = 2.25

Step 3: Solve for d

To solve for d, we can divide both sides by 0.15:

d = 2.25 / 0.15 d = 15

Step 4: Solve for q

Now that we have the value of d, we can substitute it into Equation 2:

q + 15 = 33 q = 33 - 15 q = 18

Conclusion

In this article, we set up a system of linear equations to find the number of quarters and dimes Martin has. We used the elimination method to solve the system of equations and found that Martin has 18 quarters and 15 dimes.

Key Takeaways

A system of linear equations can be used to find the number of quarters and dimes Martin has.
The total value of the coins is $6, and the total number of coins is 33.
We used the elimination method to solve the system of equations and found that Martin has 18 quarters and 15 dimes.

Real-World Applications

This problem can be applied to real-world scenarios, such as:

A store owner wants to know the number of quarters and dimes in a jar of coins.
A customer wants to know the number of quarters and dimes in a piggy bank.
A math teacher wants to create a problem for students to practice solving systems of linear equations.

Future Directions

In the future, we can explore more complex systems of linear equations and solve them using different methods, such as substitution or graphing. We can also apply these methods to real-world problems, such as finance, economics, or engineering.

References

[1] "Linear Equations" by Khan Academy
[2] "Systems of Linear Equations" by Mathway
[3] "Algebra" by OpenStax

Glossary

Linear Equation: An equation in which the highest power of the variable(s) is 1.
System of Linear Equations: A set of two or more linear equations that are solved simultaneously.
Elimination Method: A method of solving a system of linear equations by eliminating one variable and solving for the other variable.
Substitution Method: A method of solving a system of linear equations by substituting one equation into another equation and solving for the variable.

Introduction

In our previous article, we explored a system of linear equations to find the number of quarters and dimes Martin has. We used the elimination method to solve the system of equations and found that Martin has 18 quarters and 15 dimes. In this article, we will answer some frequently asked questions related to solving a system of linear equations.

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously. Each equation in the system is a linear equation, which means that the highest power of the variable(s) is 1.

Q: How do I know which method to use to solve a system of linear equations?

A: There are several methods to solve a system of linear equations, including the elimination method, substitution method, and graphing method. The choice of method depends on the type of system and the variables involved. For example, if the system has two variables and two equations, the elimination method may be the most efficient way to solve it.

Q: What is the elimination method?

A: The elimination method is a method of solving a system of linear equations by eliminating one variable and solving for the other variable. This is done by multiplying one or both equations by a constant and then adding or subtracting the equations to eliminate the variable.

Q: What is the substitution method?

A: The substitution method is a method of solving a system of linear equations by substituting one equation into another equation and solving for the variable. This is done by solving one equation for one variable and then substituting that expression into the other equation.

Q: How do I know if a system of linear equations has a unique solution, no solution, or infinitely many solutions?

A: To determine the number of solutions to a system of linear equations, you can use the following criteria:

If the system has a unique solution, the two equations must be consistent and independent.
If the system has no solution, the two equations must be inconsistent.
If the system has infinitely many solutions, the two equations must be consistent and dependent.

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

A: Yes, you can use a graphing calculator to solve a system of linear equations. Graphing calculators can be used to graph the equations and find the point of intersection, which represents the solution to the system.

Q: How do I check my work when solving a system of linear equations?

A: To check your work, you can use the following steps:

Plug the solution back into both equations to make sure it satisfies both equations.
Check that the solution is consistent with the original problem.
Use a graphing calculator to graph the equations and verify that the solution is the point of intersection.

Q: Can I use a system of linear equations to solve real-world problems?

A: Yes, you can use a system of linear equations to solve real-world problems. Systems of linear equations can be used to model a wide range of problems, including finance, economics, engineering, and more.

Q: What are some common applications of systems of linear equations?

A: Some common applications of systems of linear equations include:

Finance: Systems of linear equations can be used to model investment portfolios, budgeting, and financial planning.
Economics: Systems of linear equations can be used to model supply and demand, cost-benefit analysis, and economic forecasting.
Engineering: Systems of linear equations can be used to model mechanical systems, electrical systems, and other types of engineering problems.