Finding the Linear Combination

Oct 13, 2025 by windailys.com 115 views

Introduction

When dealing with systems of linear equations, it's essential to understand the concept of linear combinations and how they relate to the system's solution. A linear combination of two equations is a new equation formed by adding or subtracting the two original equations. In this article, we will explore how to determine which equation represents a linear combination of the given system of equations.

The System of Equations

The system of equations provided is:

{ \begin{cases} \frac{2}{3}x + \frac{5}{2}y = 15 \\ 4x + 15y = 12 \end{cases} \}

This system consists of two linear equations with two variables, x and y. To determine which equation represents a linear combination of the system, we need to examine the coefficients of the two equations.

Linear Combinations

A linear combination of two equations is a new equation formed by adding or subtracting the two original equations. This can be represented as:

{ \begin{cases} a(\frac{2}{3}x + \frac{5}{2}y) + b(4x + 15y) = a(15) + b(12) \end{cases} \}

where a and b are constants.

Finding the Linear Combination

To find the linear combination of the given system, we need to determine the values of a and b. We can do this by examining the coefficients of the two equations.

The first equation has coefficients of $\frac{2}{3}$ and $\frac{5}{2}$ , while the second equation has coefficients of 4 and 15. To form a linear combination, we need to find a combination of these coefficients that results in a new equation.

Option A: $\frac{4}{3}x = 42$

Option A is:

{ \frac{4}{3}x = 42 }

This equation is not a linear combination of the given system, as it does not involve both variables x and y.

Option B: $0 = 0$

Option B is:

{ 0 = 0 }

This equation is a linear combination of the given system, as it can be formed by adding or subtracting the two original equations.

Conclusion

In conclusion, the equation that represents a linear combination of the given system of equations is Option B: $0 = 0$ . This equation can be formed by adding or subtracting the two original equations, and it does not have a solution.

Understanding Linear Combinations

Linear combinations are an essential concept in linear algebra, and they play a crucial role in solving systems of linear equations. By understanding how to form linear combinations, we can determine which equations represent a linear combination of the system and which do not.

Real-World Applications

Linear combinations have numerous real-world applications, including:

Physics: Linear combinations are used to describe the motion of objects in physics.
Engineering: Linear combinations are used to design and optimize systems in engineering.
Computer Science: Linear combinations are used in computer science to solve problems in machine learning and data analysis.

Final Thoughts

In conclusion, the system of equations provided has no solution. However, by understanding linear combinations, we can determine which equation represents a linear combination of the system. This concept is essential in linear algebra and has numerous real-world applications.

References

Linear Algebra and Its Applications by Gilbert Strang
Introduction to Linear Algebra by Jim Hefferon
Linear Combinations and Systems of Linear Equations by Khan Academy

Glossary

Linear Combination: A new equation formed by adding or subtracting two original equations.
System of Linear Equations: A set of two or more linear equations with two or more variables.
Linear Algebra: A branch of mathematics that deals with the study of linear equations and linear transformations.

FAQs

Q: What is a linear combination? A: A linear combination is a new equation formed by adding or subtracting two original equations.
Q: How do I determine which equation represents a linear combination of the system? A: To determine which equation represents a linear combination of the system, examine the coefficients of the two equations and find a combination of these coefficients that results in a new equation.
Q: What are the real-world applications of linear combinations? A: Linear combinations have numerous real-world applications, including physics, engineering, and computer science.

Introduction

Linear combinations and systems of linear equations are fundamental concepts in linear algebra. In our previous article, we explored the concept of linear combinations and how they relate to the system's solution. In this article, we will answer some frequently asked questions about linear combinations and systems of linear equations.

Q: What is a linear combination?

A: A linear combination is a new equation formed by adding or subtracting two original equations. It is a way to combine two or more equations to form a new equation.

Q: How do I determine which equation represents a linear combination of the system?

A: To determine which equation represents a linear combination of the system, examine the coefficients of the two equations and find a combination of these coefficients that results in a new equation. You can also use the method of elimination to find the linear combination.

Q: What is the difference between a linear combination and a linear equation?

A: A linear equation is a single equation with two or more variables, while a linear combination is a new equation formed by adding or subtracting two or more original equations.

Q: Can a linear combination have a solution?

A: Yes, a linear combination can have a solution. However, if the linear combination is formed by adding or subtracting two equations that are inconsistent, the resulting equation will have no solution.

Q: How do I find the linear combination of two equations?

A: To find the linear combination of two equations, you can use the method of elimination. Multiply the two equations by necessary multiples such that the coefficients of one of the variables (x or y) are the same in both equations. Then, add or subtract the two equations to eliminate the variable.

Q: What are the real-world applications of linear combinations?

A: Linear combinations have numerous real-world applications, including physics, engineering, and computer science. They are used to describe the motion of objects, design and optimize systems, and solve problems in machine learning and data analysis.

Q: Can a linear combination be used to solve a system of linear equations?

A: Yes, a linear combination can be used to solve a system of linear equations. By finding the linear combination of the two equations, you can eliminate one of the variables and solve for the other variable.

Q: How do I know if a linear combination is consistent or inconsistent?

A: To determine if a linear combination is consistent or inconsistent, examine the coefficients of the two equations and the constant term. If the coefficients are the same and the constant term is the same, the linear combination is consistent. If the coefficients are different or the constant term is different, the linear combination is inconsistent.

Q: Can a linear combination be used to find the solution of a system of linear equations?

A: Yes, a linear combination can be used to find the solution of a system of linear equations. By finding the linear combination of the two equations, you can eliminate one of the variables and solve for the other variable.

Q: How do I use a linear combination to solve a system of linear equations?

A: To use a linear combination to solve a system of linear equations, follow these steps:

Find the linear combination of the two equations.
Eliminate one of the variables by adding or subtracting the two equations.
Solve for the other variable.
Check the solution by substituting the values back into the original equations.

Conclusion

In conclusion, linear combinations and systems of linear equations are fundamental concepts in linear algebra. By understanding how to form linear combinations and how to use them to solve systems of linear equations, you can apply these concepts to real-world problems in physics, engineering, and computer science.

References

Linear Algebra and Its Applications by Gilbert Strang
Introduction to Linear Algebra by Jim Hefferon
Linear Combinations and Systems of Linear Equations by Khan Academy

Glossary

Linear Combination: A new equation formed by adding or subtracting two original equations.
System of Linear Equations: A set of two or more linear equations with two or more variables.
Linear Algebra: A branch of mathematics that deals with the study of linear equations and linear transformations.

FAQs

Q: What is a linear combination? A: A linear combination is a new equation formed by adding or subtracting two original equations.
Q: How do I determine which equation represents a linear combination of the system? A: To determine which equation represents a linear combination of the system, examine the coefficients of the two equations and find a combination of these coefficients that results in a new equation.
Q: What are the real-world applications of linear combinations? A: Linear combinations have numerous real-world applications, including physics, engineering, and computer science.