Find The Solution Of The System Of Equations.${ \begin{array}{l} -4x + 2y = -16 \ -8x + 8y = 8 \end{array} } P R O V I D E Y O U R A N S W E R I N T H E F O R M A T : Provide Your Answer In The Format: P Ro V I D Eyo U R An S W Er In T H E F Or Ma T : { (\square, \square) \}
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Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables.
The System of Equations
The system of equations we will be solving is:
{ \begin{array}{l} -4x + 2y = -16 \\ -8x + 8y = 8 \end{array} \}
Method of Elimination
One of the most common methods for solving a system of linear equations is the method of elimination. This method involves adding or subtracting the equations in the system to eliminate one of the variables.
Step 1: Multiply the Equations by Necessary Multiples
To eliminate one of the variables, we need to make the coefficients of either x or y the same in both equations. We can do this by multiplying the equations by necessary multiples.
Let's multiply the first equation by 2 and the second equation by 1:
{ \begin{array}{l} -8x + 4y = -32 \\ -8x + 8y = 8 \end{array} \}
Step 2: Subtract the Second Equation from the First Equation
Now that the coefficients of x are the same in both equations, we can subtract the second equation from the first equation to eliminate x.
{ \begin{array}{l} -8x + 4y - (-8x + 8y) = -32 - 8 \\ 0 - 4y = -40 \\ -4y = -40 \end{array} \}
Step 3: Solve for y
Now that we have eliminated x, we can solve for y by dividing both sides of the equation by -4.
{ \begin{array}{l} -4y = -40 \\ y = \frac{-40}{-4} \\ y = 10 \end{array} \}
Step 4: Substitute y into One of the Original Equations
Now that we have found the value of y, we can substitute it into one of the original equations to find the value of x.
Let's substitute y = 10 into the first original equation:
{ \begin{array}{l} -4x + 2(10) = -16 \\ -4x + 20 = -16 \\ -4x = -36 \\ x = \frac{-36}{-4} \\ x = 9 \end{array} \}
Conclusion
In this article, we have solved a system of two linear equations with two variables using the method of elimination. We have found the values of x and y that satisfy both equations in the system.
The solution to the system of equations is:
{ (x, y) = (9, 10) \}
Discussion
Solving a system of linear equations is an important skill in mathematics and has many real-world applications. In this article, we have used the method of elimination to solve a system of two linear equations with two variables.
However, there are other methods for solving a system of linear equations, such as the method of substitution and the method of matrices. These methods can be used to solve systems of linear equations with more than two variables.
Future Work
In the future, we can explore other methods for solving a system of linear equations, such as the method of substitution and the method of matrices. We can also explore the applications of solving a system of linear equations in real-world problems.
References
- [1] "Linear Algebra and Its Applications" by Gilbert Strang
- [2] "Introduction to Linear Algebra" by Jim Hefferon
Glossary
- System of Linear Equations: A set of two or more linear equations that involve the same set of variables.
- Method of Elimination: A method for solving a system of linear equations by adding or subtracting the equations to eliminate one of the variables.
- Method of Substitution: A method for solving a system of linear equations by substituting the value of one variable into one of the equations.
- Method of Matrices: A method for solving a system of linear equations by representing the system as a matrix and using matrix operations to solve the system.
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Introduction
In our previous article, we discussed how to solve a system of linear equations using the method of elimination. However, we know that there are many more questions and doubts that readers may have. In this article, we will address some of the most frequently asked questions about solving a system of linear equations.
Q&A
Q: What is a system of linear equations?
A: A system of linear equations is a set of two or more linear equations that involve the same set of variables. For example:
{ \begin{array}{l} 2x + 3y = 7 \\ x - 2y = -3 \end{array} \}
Q: What are the different methods for solving a system of linear equations?
A: There are several methods for solving a system of linear equations, including:
- Method of Elimination: This method involves adding or subtracting the equations to eliminate one of the variables.
- Method of Substitution: This method involves substituting the value of one variable into one of the equations.
- Method of Matrices: This method involves representing the system as a matrix and using matrix operations to solve the system.
Q: How do I choose the method of elimination?
A: To choose the method of elimination, you need to look at the coefficients of the variables in the equations. If the coefficients of one variable are the same in both equations, you can eliminate that variable by subtracting the equations. If the coefficients of one variable are not the same in both equations, you can multiply one or both equations by a constant to make the coefficients the same.
Q: What is the difference between the method of elimination and the method of substitution?
A: The method of elimination involves adding or subtracting the equations to eliminate one of the variables, while the method of substitution involves substituting the value of one variable into one of the equations. The method of elimination is often faster and easier to use, but the method of substitution can be more useful when the equations are not easily simplified.
Q: Can I use the method of matrices to solve a system of linear equations?
A: Yes, you can use the method of matrices to solve a system of linear equations. This method involves representing the system as a matrix and using matrix operations to solve the system. The method of matrices is often more efficient and easier to use than the method of elimination or the method of substitution.
Q: How do I represent a system of linear equations as a matrix?
A: To represent a system of linear equations as a matrix, you need to create a matrix with the coefficients of the variables in the equations. For example, if we have the system of equations:
{ \begin{array}{l} 2x + 3y = 7 \\ x - 2y = -3 \end{array} \}
We can represent this system as the matrix:
{ \begin{bmatrix} 2 & 3 \\ 1 & -2 \end{bmatrix} \}
Q: What are some common mistakes to avoid when solving a system of linear equations?
A: Some common mistakes to avoid when solving a system of linear equations include:
- Not checking the solution: Make sure to check the solution to the system of equations to ensure that it satisfies both equations.
- Not using the correct method: Choose the correct method for solving the system of equations, such as the method of elimination or the method of substitution.
- Not simplifying the equations: Simplify the equations before solving the system to make it easier to solve.
Conclusion
Solving a system of linear equations is an important skill in mathematics and has many real-world applications. In this article, we have addressed some of the most frequently asked questions about solving a system of linear equations. We hope that this article has been helpful in clarifying any doubts or questions you may have had.
Discussion
Solving a system of linear equations is an important topic in mathematics, and there are many different methods for solving these systems. In this article, we have discussed some of the most common methods, including the method of elimination, the method of substitution, and the method of matrices.
However, there are many other methods for solving a system of linear equations, and the choice of method will depend on the specific system of equations and the preferences of the solver.
Future Work
In the future, we can explore other methods for solving a system of linear equations, such as the method of Gaussian elimination and the method of LU decomposition. We can also explore the applications of solving a system of linear equations in real-world problems.
References
- [1] "Linear Algebra and Its Applications" by Gilbert Strang
- [2] "Introduction to Linear Algebra" by Jim Hefferon
Glossary
- System of Linear Equations: A set of two or more linear equations that involve the same set of variables.
- Method of Elimination: A method for solving a system of linear equations by adding or subtracting the equations to eliminate one of the variables.
- Method of Substitution: A method for solving a system of linear equations by substituting the value of one variable into one of the equations.
- Method of Matrices: A method for solving a system of linear equations by representing the system as a matrix and using matrix operations to solve the system.