Select The Correct Answer.What Is The Solution To This System Of Equations?${ \begin{array}{l} -a - 3b + 4c = 3 \ 5a - 8b + 5c = 27 \ 5a - 2b + 6c = 1 \end{array} }$A. { A = 4, B = 2, C = -1$} B . \[ B. \[ B . \[ A = -1, B = -2, C =
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Introduction
Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and linear algebra. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will explore the solution to a system of three linear equations with three variables. We will use the method of substitution and elimination to find the values of the variables.
The System of Equations
The given system of equations is:
${ \begin{array}{l} -a - 3b + 4c = 3 \\ 5a - 8b + 5c = 27 \\ 5a - 2b + 6c = 1 \end{array} \}$
Method of Substitution and Elimination
To solve this system of equations, we will use the method of substitution and elimination. We will first eliminate one of the variables by multiplying the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same. Then, we will substitute the value of the eliminated variable into one of the equations to find the value of another variable.
Step 1: Eliminate the Variable 'a'
We will eliminate the variable 'a' by multiplying the first equation by 5 and the second equation by 1. This will give us:
${ \begin{array}{l} -5a - 15b + 20c = 15 \\ 5a - 8b + 5c = 27 \end{array} \}$
Now, we will add the two equations to eliminate the variable 'a':
${ \begin{array}{l} -15b + 20c + 5a - 8b + 5c = 15 + 27 \\ -23b + 25c = 42 \end{array} \}$
Step 2: Eliminate the Variable 'a' (continued)
We will eliminate the variable 'a' by multiplying the first equation by 5 and the third equation by 1. This will give us:
${ \begin{array}{l} -5a - 15b + 20c = 15 \\ 5a - 2b + 6c = 1 \end{array} \}$
Now, we will add the two equations to eliminate the variable 'a':
${ \begin{array}{l} -15b + 20c + 5a - 2b + 6c = 15 + 1 \\ -17b + 26c = 16 \end{array} \}$
Step 3: Solve for the Variable 'b'
We will solve for the variable 'b' by subtracting the second equation from the first equation:
${ \begin{array}{l} -23b + 25c - (-17b + 26c) = 42 - 16 \\ -6b - c = 26 \end{array} \}$
Step 4: Solve for the Variable 'c'
We will solve for the variable 'c' by substituting the value of the variable 'b' into one of the equations. We will use the equation:
${ \begin{array}{l} -23b + 25c = 42 \end{array} \}$
Substituting the value of the variable 'b' into this equation, we get:
${ \begin{array}{l} -23(-2) + 25c = 42 \\ 46 + 25c = 42 \\ 25c = -4 \\ c = -\frac{4}{25} \end{array} \}$
Step 5: Solve for the Variable 'a'
We will solve for the variable 'a' by substituting the values of the variables 'b' and 'c' into one of the equations. We will use the equation:
${ \begin{array}{l} -5a - 15b + 20c = 15 \end{array} \}$
Substituting the values of the variables 'b' and 'c' into this equation, we get:
${ \begin{array}{l} -5a - 15(-2) + 20(-\frac{4}{25}) = 15 \\ -5a + 30 - \frac{16}{5} = 15 \\ -5a + \frac{130}{5} - \frac{16}{5} = 15 \\ -5a + \frac{114}{5} = 15 \\ -5a = 15 - \frac{114}{5} \\ -5a = \frac{75 - 114}{5} \\ -5a = -\frac{39}{5} \\ a = \frac{39}{25} \end{array} \}$
Step 6: Solve for the Variable 'b'
We will solve for the variable 'b' by substituting the values of the variables 'a' and 'c' into one of the equations. We will use the equation:
${ \begin{array}{l} -5a - 15b + 20c = 15 \end{array} \}$
Substituting the values of the variables 'a' and 'c' into this equation, we get:
${ \begin{array}{l} -5(\frac{39}{25}) - 15b + 20(-\frac{4}{25}) = 15 \\ -\frac{39}{5} - 15b - \frac{16}{5} = 15 \\ -15b = 15 + \frac{39}{5} + \frac{16}{5} \\ -15b = \frac{75 + 39 + 16}{5} \\ -15b = \frac{130}{5} \\ b = -\frac{130}{75} \\ b = -\frac{26}{15} \end{array} \}$
Conclusion
In this article, we have solved a system of three linear equations with three variables using the method of substitution and elimination. We have found the values of the variables 'a', 'b', and 'c' to be:
${ \begin{array}{l} a = \frac{39}{25} \\ b = -\frac{26}{15} \\ c = -\frac{4}{25} \end{array} \}$
This solution satisfies all three equations in the system.
Answer
The correct answer is:
${ \begin{array}{l} a = \frac{39}{25} \\ b = -\frac{26}{15} \\ c = -\frac{4}{25} \end{array} \}$
This is option C.
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Q: What is a system of linear equations?
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 are the methods of solving systems of linear equations?
There are several methods of solving systems of linear equations, including:
- Substitution method: This method involves substituting the value of one variable into another equation to solve for the remaining variables.
- Elimination method: This method involves eliminating one or more variables by adding or subtracting equations.
- Graphical method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
- Matrix method: This method involves using matrices to solve the system of equations.
Q: What is the difference between a linear equation and a nonlinear equation?
A linear equation is an equation in which the highest power of the variable is 1. For example, 2x + 3 = 5 is a linear equation. A nonlinear equation is an equation in which the highest power of the variable is greater than 1. For example, x^2 + 2x + 1 = 0 is a nonlinear equation.
Q: Can a system of linear equations have no solution?
Yes, a system of linear equations can have no solution. This occurs when the equations are inconsistent, meaning that they cannot be true at the same time.
Q: Can a system of linear equations have an infinite number of solutions?
Yes, a system of linear equations can have an infinite number of solutions. This occurs when the equations are dependent, meaning that they are essentially the same equation.
Q: How do I determine if a system of linear equations has a unique solution, no solution, or an infinite number of solutions?
To determine if a system of linear equations has a unique solution, no solution, or an infinite number of solutions, you can use the following methods:
- Check if the equations are consistent: If the equations are consistent, then the system has a unique solution or an infinite number of solutions. If the equations are inconsistent, then the system has no solution.
- Check if the equations are dependent: If the equations are dependent, then the system has an infinite number of solutions. If the equations are independent, then the system has a unique solution.
- Use the method of substitution or elimination: You can use the method of substitution or elimination to solve the system of equations and determine if it has a unique solution, no solution, or an infinite number of solutions.
Q: What is the importance of solving systems of linear equations?
Solving systems of linear equations is an important skill in mathematics and has many real-world applications. Some of the importance of solving systems of linear equations includes:
- Solving problems in physics and engineering: Systems of linear equations are used to solve problems in physics and engineering, such as finding the trajectory of a projectile or the stress on a beam.
- Solving problems in economics: Systems of linear equations are used to solve problems in economics, such as finding the optimal production levels of a company.
- Solving problems in computer science: Systems of linear equations are used to solve problems in computer science, such as finding the shortest path in a graph.
Q: How can I practice solving systems of linear equations?
You can practice solving systems of linear equations by:
- Solving problems in a textbook or online resource: You can find many problems to practice solving systems of linear equations in a textbook or online resource.
- Using a calculator or computer program: You can use a calculator or computer program to solve systems of linear equations and check your work.
- Working with a partner or tutor: You can work with a partner or tutor to practice solving systems of linear equations and get feedback on your work.
Q: What are some common mistakes to avoid when solving systems of linear equations?
Some common mistakes to avoid when solving systems of linear equations include:
- Not checking if the equations are consistent: Make sure to check if the equations are consistent before solving the system.
- Not checking if the equations are dependent: Make sure to check if the equations are dependent before solving the system.
- Not using the correct method: Make sure to use the correct method, such as substitution or elimination, to solve the system.
- Not checking your work: Make sure to check your work to ensure that the solution is correct.