Lydia Buys 5 Pounds Of Apples And 3 Pounds Of Bananas For A Total Of $ 8.50 \$8.50 $8.50 . Ari Buys 3 Pounds Of Apples And 2 Pounds Of Bananas For A Total Of $ 5.25 \$5.25 $5.25 . This System Of Equations Represents The Situation, Where X X X Is The

Introduction

In this article, we will explore a real-world problem that can be represented by a system of equations. We will use the problem of buying fruits to create a system of equations and then solve it using algebraic methods. This problem is a great example of how mathematics can be applied to real-world situations.

The Problem

Lydia buys 5 pounds of apples and 3 pounds of bananas for a total of $\$8.50$. Ari buys 3 pounds of apples and 2 pounds of bananas for a total of $\$5.25$. This situation can be represented by a system of equations, where $x$ is the price of one pound of apples and $y$ is the price of one pound of bananas.

The System of Equations

Let's represent the situation as a system of equations. We can write two equations based on the information given:

• Lydia's purchase: $5x + 3y = 8.50$
• Ari's purchase: $3x + 2y = 5.25$

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

First, we will multiply the two equations by necessary multiples such that the coefficients of $y$'s in both equations are the same:

• Multiply the first equation by 2: $10x + 6y = 17$
• Multiply the second equation by 3: $9x + 6y = 15.75$

Step 2: Subtract the Equations

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

$10x + 6y - (9x + 6y) = 17 - 15.75$

Simplifying the equation, we get:

$x = 1.25$

Step 3: Substitute the Value of $x$ into One of the Original Equations

Now that we have 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:

$5x + 3y = 8.50$

Substituting $x = 1.25$, we get:

$5(1.25) + 3y = 8.50$

Simplifying the equation, we get:

$6.25 + 3y = 8.50$

Subtracting 6.25 from both sides, we get:

$3y = 2.25$

Dividing both sides by 3, we get:

$y = 0.75$

Conclusion

In this article, we solved a system of equations that represented a real-world problem. We used the elimination method to find the values of $x$ and $y$, which represent the price of one pound of apples and one pound of bananas, respectively. The solution to the system of equations is $x = 1.25$ and $y = 0.75$.

Real-World Applications

This problem has many real-world applications. For example, it can be used to determine the prices of fruits in a grocery store. By knowing the prices of apples and bananas, we can calculate the total cost of a purchase.

Conclusion

In conclusion, solving a system of equations is an essential skill in mathematics. It can be used to represent real-world problems and find solutions to them. By using algebraic methods, we can solve systems of equations and find the values of variables that represent the solution to the problem.

Future Work

In the future, we can explore more real-world problems that can be represented by systems of equations. We can use different algebraic methods to solve the systems of equations and find the solutions to the problems.

References

• [1] "Systems of Equations". Khan Academy.
• [2] "Solving Systems of Equations". Mathway.

Glossary

• System of Equations: A set of two or more equations that are related to each other.
• Elimination Method: A method of solving a system of equations by eliminating one of the variables.
• Substitution Method: A method of solving a system of equations by substituting one of the variables into the other equation.
  Frequently Asked Questions (FAQs) About Solving Systems of Equations ====================================================================

Introduction

In our previous article, we solved a system of equations that represented a real-world problem. We used the elimination method to find the values of $x$ and $y$, which represent the price of one pound of apples and one pound of bananas, respectively. In this article, we will answer some frequently asked questions (FAQs) about solving systems of equations.

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. Each equation in the system is a statement that two expressions are equal.

Q: How do I know if I have a system of equations?

A: You have a system of equations if you have two or more equations that are related to each other. For example, if you have two equations like $x + y = 3$ and $2x + 2y = 6$, you have a system of equations.

Q: What are the different methods of solving systems of equations?

A: There are two main methods of solving systems of equations: the elimination method and the substitution method.

• Elimination Method: This method involves eliminating one of the variables by adding or subtracting the equations.
• Substitution Method: This method involves substituting one of the variables into the other equation.

Q: How do I choose which method to use?

A: You can choose which method to use based on the coefficients of the variables in the equations. If the coefficients of the variables are the same, you can use the elimination method. If the coefficients of the variables are different, you can use the substitution method.

Q: What is the elimination method?

A: The elimination method is a method of solving systems of equations by eliminating one of the variables. This is done by adding or subtracting the equations.

Q: How do I use the elimination method?

A: To use the elimination method, follow these steps:

1. Multiply the equations by necessary multiples such that the coefficients of the variables are the same.
2. Add or subtract the equations to eliminate one of the variables.
3. Solve for the remaining variable.
4. Substitute the value of the remaining variable into one of the original equations to find the value of the other variable.

Q: What is the substitution method?

A: The substitution method is a method of solving systems of equations by substituting one of the variables into the other equation.

Q: How do I use the substitution method?

A: To use the substitution method, follow these steps:

1. Solve one of the equations for one of the variables.
2. Substitute the value of the variable into the other equation.
3. Solve for the remaining variable.
4. Substitute the value of the remaining variable into one of the original equations to find the value of the other variable.

Q: What are some common mistakes to avoid when solving systems of equations?

A: Some common mistakes to avoid when solving systems of equations include:

• Not checking if the equations are consistent.
• Not checking if the equations are independent.
• Not using the correct method to solve the system of equations.
• Not checking if the solution is valid.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about solving systems of equations. We discussed the different methods of solving systems of equations, including the elimination method and the substitution method. We also discussed some common mistakes to avoid when solving systems of equations.

Real-World Applications

Solving systems of equations has many real-world applications. For example, it can be used to determine the prices of fruits in a grocery store. By knowing the prices of apples and bananas, we can calculate the total cost of a purchase.

Conclusion

In conclusion, solving systems of equations is an essential skill in mathematics. It can be used to represent real-world problems and find solutions to them. By using algebraic methods, we can solve systems of equations and find the values of variables that represent the solution to the problem.

Future Work

In the future, we can explore more real-world problems that can be represented by systems of equations. We can use different algebraic methods to solve the systems of equations and find the solutions to the problems.

References

• [1] "Systems of Equations". Khan Academy.
• [2] "Solving Systems of Equations". Mathway.

Glossary

• System of Equations: A set of two or more equations that are related to each other.
• Elimination Method: A method of solving a system of equations by eliminating one of the variables.
• Substitution Method: A method of solving a system of equations by substituting one of the variables into the other equation.