Henrique Began To Solve A System Of Linear Equations Using The Linear Combination Method. His Work Is Shown Below:1. $[ \begin{array}{l} 3(4x - 7y = 28) \quad \rightarrow \quad 12x - 21y = 84 \ -2(6x - 5y = 31) \quad \rightarrow \quad -12x + 10y

Introduction

Solving systems of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. In this article, we will explore the linear combination method, a popular technique used to solve systems of linear equations. We will analyze the work of Henrique, a student who is attempting to solve a system of linear equations using the linear combination method.

The Linear Combination Method

The linear combination method is a technique used to solve systems of linear equations by combining two or more equations to eliminate one of the variables. This method is based on the concept of linear combinations, which states that any linear combination of two or more equations can be used to eliminate one of the variables.

Henrique's Work

Henrique is attempting to solve the following system of linear equations using the linear combination method:

1. $4x - 7y = 28$
2. $6x - 5y = 31$

Henrique's work is shown below:

1. {

\begin{array}{l} 3(4x - 7y = 28) \quad \rightarrow \quad 12x - 21y = 84 \ -2(6x - 5y = 31) \quad \rightarrow \quad -12x + 10y \end{array} }$

Analyzing Henrique's Work

Let's analyze Henrique's work and identify the mistakes he made.

1. Henrique multiplied the first equation by 3, which resulted in the equation $12x - 21y = 84$. This is a correct step.
2. Henrique multiplied the second equation by -2, which resulted in the equation $-12x + 10y = -62$. This is also a correct step.
3. However, Henrique failed to combine the two equations to eliminate one of the variables. He simply listed the two equations side by side, without combining them.

Correcting Henrique's Work

To correct Henrique's work, we need to combine the two equations to eliminate one of the variables. Let's try to eliminate the variable x.

We can add the two equations to eliminate the variable x:

$12x - 21y = 84$

$-12x + 10y = -62$

Adding the two equations, we get:

$-11y = 22$

Now, we can solve for y by dividing both sides by -11:

$y = -2$

Solving for x

Now that we have the value of y, we can substitute it into one of the original equations to solve for x. Let's use the first equation:

$4x - 7y = 28$

Substituting y = -2, we get:

$4x - 7(-2) = 28$

Simplifying the equation, we get:

$4x + 14 = 28$

Subtracting 14 from both sides, we get:

$4x = 14$

Dividing both sides by 4, we get:

$x = \frac{14}{4}$

$x = \frac{7}{2}$

Conclusion

In this article, we analyzed Henrique's work and identified the mistakes he made while attempting to solve a system of linear equations using the linear combination method. We corrected his work by combining the two equations to eliminate one of the variables and solved for x and y. The linear combination method is a powerful technique used to solve systems of linear equations, and it has numerous applications in various fields.

Tips and Tricks

Here are some tips and tricks to help you solve systems of linear equations using the linear combination method:

• Make sure to multiply the equations by the correct coefficients to eliminate one of the variables.
• Combine the equations to eliminate one of the variables.
• Solve for the variable that is eliminated.
• Substitute the value of the eliminated variable into one of the original equations to solve for the other variable.
• Check your work by plugging the values of x and y back into the original equations.

Common Mistakes

Here are some common mistakes to avoid when solving systems of linear equations using the linear combination method:

• Failing to multiply the equations by the correct coefficients.
• Failing to combine the equations to eliminate one of the variables.
• Solving for the wrong variable.
• Failing to substitute the value of the eliminated variable into one of the original equations.
• Failing to check your work by plugging the values of x and y back into the original equations.

Real-World Applications

The linear combination method has numerous real-world applications in various fields such as physics, engineering, economics, and computer science. Here are some examples:

• Physics: The linear combination method is used to solve systems of linear equations that describe the motion of objects in physics.
• Engineering: The linear combination method is used to solve systems of linear equations that describe the behavior of electrical circuits and mechanical systems.
• Economics: The linear combination method is used to solve systems of linear equations that describe the behavior of economic systems and markets.
• Computer Science: The linear combination method is used to solve systems of linear equations that describe the behavior of computer networks and algorithms.

Conclusion

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 to find the values of the variables.

Q: What is the linear combination method?

A: The linear combination method is a technique used to solve systems of linear equations by combining two or more equations to eliminate one of the variables.

Q: How do I know which variable to eliminate?

A: To determine which variable to eliminate, you need to look at the coefficients of the variables in the two equations. If the coefficients of one variable are the same in both equations, you can eliminate that variable.

Q: What is the difference between the linear combination method and the substitution method?

A: The linear combination method involves combining two or more equations to eliminate one of the variables, while the substitution method involves substituting the value of one variable into one of the equations to solve for the other variable.

Q: Can I use the linear combination method to solve a system of linear equations with three or more variables?

A: Yes, you can use the linear combination method to solve a system of linear equations with three or more variables. However, you need to be careful to eliminate the correct variable and to follow the correct steps.

Q: What are some common mistakes to avoid when using the linear combination method?

A: Some common mistakes to avoid when using the linear combination method include:

• Failing to multiply the equations by the correct coefficients.
• Failing to combine the equations to eliminate one of the variables.
• Solving for the wrong variable.
• Failing to substitute the value of the eliminated variable into one of the original equations.
• Failing to check your work by plugging the values of x and y back into the original equations.

Q: How do I check my work when using the linear combination method?

A: To check your work when using the linear combination method, you need to plug the values of x and y back into the original equations to make sure they are true.

Q: Can I use the linear combination method to solve a system of linear equations with fractions?

A: Yes, you can use the linear combination method to solve a system of linear equations with fractions. However, you need to be careful to multiply the equations by the correct coefficients to eliminate the fractions.

Q: What are some real-world applications of the linear combination method?

A: Some real-world applications of the linear combination method include:

• Physics: The linear combination method is used to solve systems of linear equations that describe the motion of objects in physics.
• Engineering: The linear combination method is used to solve systems of linear equations that describe the behavior of electrical circuits and mechanical systems.
• Economics: The linear combination method is used to solve systems of linear equations that describe the behavior of economic systems and markets.
• Computer Science: The linear combination method is used to solve systems of linear equations that describe the behavior of computer networks and algorithms.

Q: How do I know if a system of linear equations has a solution?

A: To determine if a system of linear equations has a solution, you need to check if the equations are consistent. If the equations are consistent, then the system has a solution. If the equations are inconsistent, then the system does not have a solution.

Q: What is the difference between a consistent and an inconsistent system of linear equations?

A: A consistent system of linear equations is one that has a solution, while an inconsistent system of linear equations is one that does not have a solution.

Q: Can I use the linear combination method to solve a system of linear equations that is inconsistent?

A: No, you cannot use the linear combination method to solve a system of linear equations that is inconsistent. If the system is inconsistent, then it does not have a solution, and the linear combination method will not be able to find a solution.

Conclusion

In conclusion, the linear combination method is a powerful technique used to solve systems of linear equations. It has numerous real-world applications in various fields and is a fundamental concept in mathematics. By following the tips and tricks outlined in this article, you can master the linear combination method and solve systems of linear equations with ease.