Solve The System Of Equations:(a) 4 X + 3 Y = 24 3 X + Y = − 2 \begin{array}{l} 4x + 3y = 24 \\ 3x + Y = -2 \end{array} 4 X + 3 Y = 24 3 X + Y = − 2

Mar 13, 2025 by ADMIN 151 views

**Solving Systems of Linear Equations: A Comprehensive Guide**

Introduction

Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will delve into the world of systems of linear equations and provide a step-by-step guide on how to solve them. We will use the given system of equations as a case study to illustrate the concepts and techniques involved.

What are Systems of Linear Equations?

A system of linear equations is a set of two or more linear equations that involve the same variables. Each equation is a linear equation, meaning it is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables. The system of equations is said to be consistent if it has a solution, and inconsistent if it does not have a solution.

The Given System of Equations

The given system of equations is:

$\begin{array}{l} 4x + 3y = 24 \\ 3x + y = -2 \end{array}$

Method 1: Substitution Method

One way to solve this system of equations is by using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.

Step 1: Solve the Second Equation for y

We can solve the second equation for y by subtracting 3x from both sides:

$y = -2 - 3x$

Step 2: Substitute the Expression for y into the First Equation

Now, we can substitute the expression for y into the first equation:

$4x + 3(-2 - 3x) = 24$

Step 3: Simplify the Equation

Simplifying the equation, we get:

$4x - 6 - 9x = 24$

Step 4: Combine Like Terms

Combining like terms, we get:

$-5x - 6 = 24$

Step 5: Add 6 to Both Sides

Adding 6 to both sides, we get:

$-5x = 30$

Step 6: Divide Both Sides by -5

Dividing both sides by -5, we get:

$x = -6$

Step 7: Substitute the Value of x into the Expression for y

Now, we can substitute the value of x into the expression for y:

$y = -2 - 3(-6)$

Step 8: Simplify the Expression

Simplifying the expression, we get:

$y = -2 + 18$

Step 9: Combine Like Terms

Combining like terms, we get:

$y = 16$

Method 2: Elimination Method

Another way to solve this system of equations is by using the elimination method. This method involves adding or subtracting the equations to eliminate one variable.

Step 1: Multiply the First Equation by 1 and the Second Equation by 3

We can multiply the first equation by 1 and the second equation by 3 to make the coefficients of y in both equations equal:

$4x + 3y = 24$

$9x + 3y = -6$

Step 2: Subtract the First Equation from the Second Equation

Now, we can subtract the first equation from the second equation to eliminate the variable y:

$(9x - 4x) + (3y - 3y) = -6 - 24$

Step 3: Simplify the Equation

Simplifying the equation, we get:

$5x = -30$

Step 4: Divide Both Sides by 5

Dividing both sides by 5, we get:

$x = -6$

Step 5: Substitute the Value of x into the First Equation

Now, we can substitute the value of x into the first equation:

$4(-6) + 3y = 24$

Step 6: Simplify the Equation

Simplifying the equation, we get:

$-24 + 3y = 24$

Step 7: Add 24 to Both Sides

Adding 24 to both sides, we get:

$3y = 48$

Step 8: Divide Both Sides by 3

Dividing both sides by 3, we get:

$y = 16$

Conclusion

In this article, we have solved the given system of equations using two different methods: substitution and elimination. Both methods have led to the same solution: x = -6 and y = 16. We have also discussed the concept of systems of linear equations and provided a step-by-step guide on how to solve them.

Tips and Tricks

When solving systems of linear equations, it is essential to check the consistency of the system before solving it.
The substitution method is useful when one equation is already solved for one variable.
The elimination method is useful when the coefficients of one variable in both equations are equal.
When using the elimination method, make sure to multiply the equations by the correct multiples to make the coefficients of one variable equal.

Practice Problems

Solve the system of equations:

$\begin{array}{l} 2x + 3y = 12 \\ x + 2y = 6 \end{array}$

Solve the system of equations:

$\begin{array}{l} 3x + 2y = 10 \\ 2x + 3y = 8 \end{array}$

Solve the system of equations:

$\begin{array}{l} x + 2y = 4 \\ 2x + 3y = 7 \end{array}$

References

[1] "Systems of Linear Equations" by Khan Academy
[2] "Solving Systems of Linear Equations" by Math Open Reference
[3] "Systems of Linear Equations" by Wolfram MathWorld
Frequently Asked Questions (FAQs) About Systems of Linear Equations ====================================================================

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 variables. Each equation is a linear equation, meaning it is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables.

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 system is consistent. A system is consistent if it has a solution, and inconsistent if it does not have a solution. You can check the consistency of a system by using the method of substitution or elimination.

Q: What is the difference between the substitution method and the elimination method?

A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable.

Q: How do I choose between the substitution method and the elimination method?

A: You can choose between the substitution method and the elimination method based on the coefficients of the variables in the equations. If the coefficients of one variable in both equations are equal, it is easier to use the elimination method. If one equation is already solved for one variable, it is easier to use the substitution method.

Q: What is the importance of checking the consistency of a system of linear equations?

A: Checking the consistency of a system of linear equations is essential to ensure that the system has a solution. If a system is inconsistent, it means that there is no solution, and you need to re-evaluate the equations.

Q: How do I solve a system of linear equations with three variables?

A: To solve a system of linear equations with three variables, you can use the method of substitution or elimination. You can also use the method of matrices to solve the system.

Q: What is the method of matrices?

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

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

A: To use the method of matrices to solve a system of linear equations, you need to represent the system as a matrix and then perform operations on the matrix to solve the system. You can use the following steps:

Represent the system as a matrix.
Perform row operations on the matrix to get a matrix in row-echelon form.
Use the matrix in row-echelon form to solve the system.

Q: What is the row-echelon form of a matrix?

A: The row-echelon form of a matrix is a matrix in which all the entries below the leading entry in each row are zero.

Q: How do I perform row operations on a matrix?

A: To perform row operations on a matrix, you can use the following steps:

Swap two rows.
Multiply a row by a non-zero constant.
Add a multiple of one row to another row.

Q: What are the advantages of using the method of matrices to solve a system of linear equations?

A: The advantages of using the method of matrices to solve a system of linear equations include:

It is a systematic approach to solving the system.
It is easier to use than the substitution method or elimination method.
It can be used to solve systems with three or more variables.

Q: What are the disadvantages of using the method of matrices to solve a system of linear equations?

A: The disadvantages of using the method of matrices to solve a system of linear equations include:

It requires a good understanding of matrix operations.
It can be time-consuming to perform the row operations.
It may not be as intuitive as the substitution method or elimination method.

Conclusion

In this article, we have answered some frequently asked questions about systems of linear equations. We have discussed the importance of checking the consistency of a system, the difference between the substitution method and the elimination method, and the advantages and disadvantages of using the method of matrices to solve a system of linear equations. We hope that this article has been helpful in providing a better understanding of systems of linear equations.