Use The Linear Combination Method To Solve The System Of Equations:$\[ \begin{cases} 5x + Y = 2 \\ 4x + Y = 4 \end{cases} \\]Choose The Correct Solution From The Options Below:A. (0, 2)B. (-2, 12)C. (1, -8)D. (-3, 17)

Feb 27, 2025 by ADMIN 220 views

**Solving Systems of Linear Equations using the Linear Combination Method**

Introduction

In mathematics, a system of linear equations is a set of two or more equations in which the unknowns are related through linear equations. Solving systems of linear equations is a fundamental concept in algebra and is used extensively in various fields such as physics, engineering, economics, and computer science. In this article, we will focus on solving a system of two linear equations using the linear combination method.

What is the Linear Combination Method?

The linear combination method is a technique used to solve systems of linear equations by combining the equations in a way that eliminates 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.

Step 1: Write Down the System of Equations

The system of equations we will be solving is:

{ \begin{cases} 5x + y = 2 \\ 4x + y = 4 \end{cases} \}

Step 2: Subtract the Second Equation from the First Equation

To eliminate the variable y, we will subtract the second equation from the first equation. This will give us a new equation with only one variable, x.

{ (5x + y) - (4x + y) = 2 - 4 \}

Simplifying the equation, we get:

{ x = -2 \}

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 substitute x = -2 into the first equation:

{ 5(-2) + y = 2 \}

Simplifying the equation, we get:

{ -10 + y = 2 \}

Adding 10 to both sides of the equation, we get:

{ y = 12 \}

Step 4: Write Down the Solution

The solution to the system of equations is x = -2 and y = 12. Therefore, the correct solution is:

{ (-2, 12) \}

Conclusion

In this article, we used the linear combination method to solve a system of two linear equations. We first wrote down the system of equations, then subtracted the second equation from the first equation to eliminate the variable y. We then substituted the value of x into one of the original equations to find the value of y. Finally, we wrote down the solution to the system of equations.

Options

A. (0, 2) B. (-2, 12) C. (1, -8) D. (-3, 17)

The correct solution is B. (-2, 12).

Discussion

The linear combination method is a powerful technique for solving systems of linear equations. It 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. This method is used extensively in various fields such as physics, engineering, economics, and computer science.

Example Problems

Solve the system of equations using the linear combination method:

{ \begin{cases} 2x + 3y = 5 \\ x + 2y = 3 \end{cases} \}

Solve the system of equations using the linear combination method:

{ \begin{cases} x + 2y = 4 \\ 2x + 3y = 7 \end{cases} \}

Solutions

The solution to the system of equations is x = 1 and y = 1.
The solution to the system of equations is x = 2 and y = 1.

Tips and Tricks

When using the linear combination method, make sure to subtract the second equation from the first equation to eliminate the variable y.
When substituting the value of x into one of the original equations, make sure to simplify the equation to find the value of y.
When writing down the solution, make sure to include both the value of x and the value of y.

Conclusion

Q: What is the linear combination method?

A: The linear combination method is a technique used to solve systems of linear equations by combining the equations in a way that eliminates one of the variables.

Q: How do I know which equation to subtract from which?

A: To eliminate one of the variables, you need to subtract the second equation from the first equation. This will give you a new equation with only one variable.

Q: What if I get a negative value for one of the variables?

A: If you get a negative value for one of the variables, it means that the value of the variable is negative. For example, if you get x = -2, it means that the value of x is -2.

Q: Can I use the linear combination method to solve systems of linear equations with more than two variables?

A: No, the linear combination method is only used to solve systems of linear equations with two variables.

Q: What if I get a fraction as a value for one of the variables?

A: If you get a fraction as a value for one of the variables, you can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor.

Q: Can I use the linear combination method to solve systems of linear equations with decimals?

A: Yes, you can use the linear combination method to solve systems of linear equations with decimals.

Q: What if I get a complex number as a value for one of the variables?

A: If you get a complex number as a value for one of the variables, you can simplify the complex number by using the formula for complex numbers.

Q: Can I use the linear combination method to solve systems of linear equations with absolute values?

A: Yes, you can use the linear combination method to solve systems of linear equations with absolute values.

Q: What if I get a system of linear equations with no solution?

A: If you get a system of linear equations with no solution, it means that the system of equations is inconsistent and has no solution.

Q: Can I use the linear combination method to solve systems of linear equations with infinitely many solutions?

A: Yes, you can use the linear combination method to solve systems of linear equations with infinitely many solutions.

Q: What if I get a system of linear equations with a single solution?

A: If you get a system of linear equations with a single solution, it means that the system of equations is consistent and has a unique solution.

Q: Can I use the linear combination method to solve systems of linear equations with parameters?

A: Yes, you can use the linear combination method to solve systems of linear equations with parameters.

Q: What if I get a system of linear equations with a parameter that is not a variable?

A: If you get a system of linear equations with a parameter that is not a variable, you can substitute the value of the parameter into the system of equations and solve for the variables.

Conclusion

In this article, we answered some frequently asked questions about solving systems of linear equations using the linear combination method. We covered topics such as how to know which equation to subtract from which, what to do if you get a negative value for one of the variables, and how to handle systems of linear equations with more than two variables. We also covered topics such as how to handle systems of linear equations with decimals, complex numbers, and absolute values.