Find The General Solution Of The Following System Of Equations.$ \begin{cases} -4x - Y = 7 \ 12x + 3y = -21 \end{cases} }$Select The Correct Answer Below A. { (x, \frac{x {4} - \frac{7}{4})$} B . \[ B. \[ B . \[ (x, \frac{x}{4} +

Mar 9, 2025 by ADMIN 231 views

**Solving a System of Linear Equations: A Step-by-Step Guide**

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Introduction

In mathematics, a system of linear equations is a set of two or more equations that involve two or more variables. Solving a system of linear equations involves 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{cases} -4x - y = 7 \\ 12x + 3y = -21 \end{cases} \}

Method of Elimination

One of the methods used to solve 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 but with opposite signs. We can do this by multiplying the equations by necessary multiples.

Let's multiply the first equation by 3 and the second equation by 1:

{ \begin{cases} -12x - 3y = 21 \\ 12x + 3y = -21 \end{cases} \}

Step 2: Add the Equations

Now that the coefficients of y are the same in both equations but with opposite signs, we can add the equations to eliminate y.

{ -12x - 3y + 12x + 3y = 21 - 21 \}

Simplifying the equation, we get:

{ 0 = 0 \}

This means that the equation is an identity, and we cannot eliminate y using this method.

Step 3: Use a Different Method

Since we cannot eliminate y using the method of elimination, we will use a different method to solve the system of equations.

Method of Substitution

Another method used to solve a system of linear equations is the method of substitution. This method involves solving one of the equations for one of the variables and then substituting that expression into the other equation.

Let's solve the first equation for y:

{ -4x - y = 7 \}

Solving for y, we get:

{ y = -4x - 7 \}

Now, let's substitute this expression for y into the second equation:

{ 12x + 3(-4x - 7) = -21 \}

Simplifying the equation, we get:

{ 12x - 12x - 21 = -21 \}

This means that the equation is an identity, and we cannot solve the system of equations using the method of substitution.

Method of Matrices

Another method used to solve a system of linear equations is the method of matrices. This method involves representing the system of equations as a matrix and then using row operations to solve the system.

Let's represent the system of equations as a matrix:

{ \begin{bmatrix} -4 & -1 \\ 12 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ -21 \end{bmatrix} \}

Row Operations

To solve the system of equations, we need to perform row operations on the matrix.

Let's multiply the first row by 3 and the second row by 1:

{ \begin{bmatrix} -12 & -3 \\ 12 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 21 \\ -21 \end{bmatrix} \}

Now, let's add the first row to the second row:

{ \begin{bmatrix} -12 & -3 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 21 \\ 0 \end{bmatrix} \}

This means that the system of equations has infinitely many solutions.

General Solution

Since the system of equations has infinitely many solutions, we can write the general solution as:

{ (x, y) = (x, \frac{x}{4} - \frac{7}{4}) \}

This means that the correct answer is:

{ (x, y) = (x, \frac{x}{4} - \frac{7}{4}) \}

The final answer is: $\boxed{A}$

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Introduction

The System of Equations

The system of equations we will be solving is:

{ \begin{cases} -4x - y = 7 \\ 12x + 3y = -21 \end{cases} \}

Method of Elimination

One of the methods used to solve 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

Let's multiply the first equation by 3 and the second equation by 1:

{ \begin{cases} -12x - 3y = 21 \\ 12x + 3y = -21 \end{cases} \}

Step 2: Add the Equations

Now that the coefficients of y are the same in both equations but with opposite signs, we can add the equations to eliminate y.

{ -12x - 3y + 12x + 3y = 21 - 21 \}

Simplifying the equation, we get:

{ 0 = 0 \}

This means that the equation is an identity, and we cannot eliminate y using this method.

Step 3: Use a Different Method

Since we cannot eliminate y using the method of elimination, we will use a different method to solve the system of equations.

Method of Substitution

Let's solve the first equation for y:

{ -4x - y = 7 \}

Solving for y, we get:

{ y = -4x - 7 \}

Now, let's substitute this expression for y into the second equation:

{ 12x + 3(-4x - 7) = -21 \}

Simplifying the equation, we get:

{ 12x - 12x - 21 = -21 \}

This means that the equation is an identity, and we cannot solve the system of equations using the method of substitution.

Method of Matrices

Let's represent the system of equations as a matrix:

{ \begin{bmatrix} -4 & -1 \\ 12 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ -21 \end{bmatrix} \}

Row Operations

To solve the system of equations, we need to perform row operations on the matrix.

Let's multiply the first row by 3 and the second row by 1:

{ \begin{bmatrix} -12 & -3 \\ 12 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 21 \\ -21 \end{bmatrix} \}

Now, let's add the first row to the second row:

{ \begin{bmatrix} -12 & -3 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 21 \\ 0 \end{bmatrix} \}

This means that the system of equations has infinitely many solutions.

General Solution

Since the system of equations has infinitely many solutions, we can write the general solution as:

{ (x, y) = (x, \frac{x}{4} - \frac{7}{4}) \}

This means that the correct answer is:

{ (x, y) = (x, \frac{x}{4} - \frac{7}{4}) \}

The final answer is: $\boxed{A}$

Q&A

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more equations that involve two or more variables.

Q: What are the methods used to solve a system of linear equations?

A: The methods used to solve a system of linear equations are the method of elimination, the method of substitution, and the method of matrices.

Q: What is the method of elimination?

A: The method of elimination involves adding or subtracting the equations in the system to eliminate one of the variables.

Q: What is the method of substitution?

A: The method of substitution involves solving one of the equations for one of the variables and then substituting that expression into the other equation.

Q: What is the method of matrices?

A: The method of matrices involves representing the system of equations as a matrix and then using row operations to solve the system.

Q: What is the general solution of a system of linear equations?

A: The general solution of a system of linear equations is the set of all possible solutions to the system.

Q: How do I know if a system of linear equations has a unique solution, infinitely many solutions, or no solution?

A: You can determine the number of solutions to a system of linear equations by looking at the number of rows and columns in the matrix representation of the system. If the number of rows is equal to the number of columns, then the system has a unique solution. If the number of rows is greater than the number of columns, then the system has infinitely many solutions. If the number of rows is less than the number of columns, then the system has no solution.

Q: How do I solve a system of linear equations using the method of elimination?

A: To solve a system of linear equations using the method of elimination, you need to add or subtract the equations in the system to eliminate one of the variables.

Q: How do I solve a system of linear equations using the method of substitution?

A: To solve a system of linear equations using the method of substitution, you need to solve one of the equations for one of the variables and then substitute that expression into the other equation.

Q: How do I solve a system of linear equations using the method of matrices?

A: To solve a system of linear equations using the method of matrices, you need to represent the system of equations as a matrix and then use row operations to solve the system.

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 if the system has a unique solution, infinitely many solutions, or no solution
Not using the correct method to solve the system
Not performing the correct row operations when using the method of matrices
Not checking if the solution satisfies all the equations in the system

Q: How do I check if a solution satisfies all the equations in the system?

A: To check if a solution satisfies all the equations in the system, you need to substitute the values of the variables into each equation and check if the equation is true.