System of Linear Equations

Oct 13, 2025 by windailys.com 137 views

Introduction

In mathematics, 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. In this article, we will focus on solving a system of two linear equations with two variables. We will use the method of substitution and elimination to find the values of the variables.

The System of Equations

The system of equations we will be solving is:

{ \begin{array}{l} -4x + 7y = -14 \\ -2x + 9y = -18 \end{array} \}

Method of Substitution

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

Let's solve the first equation for $x$ :

{ -4x + 7y = -14 \}

{ -4x = -14 - 7y \}

{ x = \frac{-14 - 7y}{-4} \}

{ x = \frac{7 + 7y}{4} \}

Now, substitute this expression for $x$ into the second equation:

{ -2x + 9y = -18 \}

{ -2\left(\frac{7 + 7y}{4}\right) + 9y = -18 \}

{ \frac{-14 - 14y}{4} + 9y = -18 \}

{ -14 - 14y + 36y = -72 \}

{ 22y = -58 \}

{ y = \frac{-58}{22} \}

{ y = -\frac{29}{11} \}

Method of Elimination

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

Let's multiply the first equation by 2 and the second equation by 4 to make the coefficients of $x$ the same:

{ -8x + 14y = -28 \}

{ -8x + 36y = -72 \}

Now, subtract the first equation from the second equation:

{ -8x + 36y - (-8x + 14y) = -72 - (-28) \}

{ 22y = -44 \}

{ y = \frac{-44}{22} \}

{ y = -\frac{22}{11} \}

Finding the Value of $x$

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 $y = -\frac{29}{11}$ into the first equation:

{ -4x + 7y = -14 \}

{ -4x + 7\left(-\frac{29}{11}\right) = -14 \}

{ -4x - \frac{203}{11} = -14 \}

{ -4x = -14 + \frac{203}{11} \}

{ -4x = \frac{-154 + 203}{11} \}

{ -4x = \frac{49}{11} \}

{ x = \frac{-49}{44} \}

Conclusion

In this article, we have solved a system of two linear equations with two variables using the method of substitution and elimination. We have found the values of $x$ and $y$ to be $x = \frac{-49}{44}$ and $y = -\frac{29}{11}$ , respectively.

Discussion

The method of substitution and elimination are two common methods used to solve systems of linear equations. The method of substitution involves solving one of the equations for one variable and then substituting that expression into the other equation. The method of elimination involves adding or subtracting the equations to eliminate one of the variables.

In this article, we have used the method of substitution and elimination to solve a system of two linear equations with two variables. We have found the values of $x$ and $y$ to be $x = \frac{-49}{44}$ and $y = -\frac{29}{11}$ , respectively.

Applications

Systems of linear equations have many applications in mathematics and other fields. Some examples include:

Linear Programming: Systems of linear equations are used to solve linear programming problems, which involve finding the optimal solution to a problem subject to certain constraints.
Computer Graphics: Systems of linear equations are used to perform transformations on objects in computer graphics, such as rotating and scaling.
Physics: Systems of linear equations are used to model physical systems, such as the motion of objects under the influence of forces.

Future Work

In the future, we can explore other methods for solving systems of linear equations, such as the method of matrices and the method of determinants. We can also apply systems of linear equations to more complex problems, such as systems of nonlinear equations and systems of differential equations.

References

"Linear Algebra and Its Applications" by Gilbert Strang
"Introduction to Linear Algebra" by Gilbert Strang
"Linear Algebra and Its Applications" by David C. Lay

Glossary

System of Linear Equations: A set of two or more linear equations that are solved simultaneously to find the values of the variables.
Method of Substitution: A method for solving systems of linear equations by solving one of the equations for one variable and then substituting that expression into the other equation.
Method of Elimination: A method for solving systems of linear equations by adding or subtracting the equations to eliminate one of the variables.
Linear Programming: A method for solving optimization problems subject to certain constraints.
Computer Graphics: The use of computers to create and manipulate visual images.
Physics: The study of the natural world around us.

Introduction

In our previous article, we solved a system of two linear equations with two variables using the method of substitution and elimination. In this article, we will answer some common questions related to solving 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 are solved simultaneously to find the values of the variables.

Q: What are the methods for solving systems of linear equations?

A: There are two common methods for solving systems of linear equations: the method of substitution and the method of elimination.

Q: What is the method of substitution?

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

Q: What is the method of elimination?

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

Q: How do I choose which method to use?

A: You can choose which method to use based on the coefficients of the variables in the equations. If the coefficients of one variable are the same, you can use the method of elimination. If the coefficients of one variable are different, you can use the method of substitution.

Q: What if I have a system of three or more linear equations?

A: If you have a system of three or more linear equations, you can use the method of substitution and elimination to solve it. You can also use the method of matrices and determinants to solve it.

Q: What if I have a system of nonlinear equations?

A: If you have a system of nonlinear equations, you cannot use the method of substitution and elimination to solve it. You will need to use other methods, such as the method of matrices and determinants, or numerical methods.

Q: What are some common applications of systems of linear equations?

A: Systems of linear equations have many applications in mathematics and other fields, including linear programming, computer graphics, and physics.

Q: How do I apply systems of linear equations to real-world problems?

A: To apply systems of linear equations to real-world problems, you need to identify the variables and the equations that describe the problem. You can then use the methods of substitution and elimination to solve the system of equations and find the values of the variables.

Q: What are some common mistakes to avoid when solving systems of linear equations?

A: Some common mistakes to avoid when solving systems of linear equations include:

Not checking for extraneous solutions: Make sure to check for extraneous solutions by plugging the values of the variables back into the original equations.
Not using the correct method: Make sure to use the correct method for solving the system of equations, based on the coefficients of the variables.
Not checking for consistency: Make sure to check for consistency by plugging the values of the variables back into the original equations.

Q: How do I check for consistency?

A: To check for consistency, you need to plug the values of the variables back into the original equations and make sure that the equations are true.

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

A: If you have a system of linear equations with no solution, it means that the equations are inconsistent. In this case, you will not be able to find a solution to the system of equations.

Q: What if I have a system of linear equations with infinitely many solutions?

A: If you have a system of linear equations with infinitely many solutions, it means that the equations are dependent. In this case, you will be able to find a solution to the system of equations, but there will be infinitely many solutions.

Conclusion

In this article, we have answered some common questions related to solving systems of linear equations. We have discussed the methods of substitution and elimination, and how to choose which method to use. We have also discussed some common applications of systems of linear equations and how to apply them to real-world problems. Finally, we have discussed some common mistakes to avoid when solving systems of linear equations.

Glossary

System of Linear Equations: A set of two or more linear equations that are solved simultaneously to find the values of the variables.
Method of Substitution: A method for solving systems of linear equations by solving one of the equations for one variable and then substituting that expression into the other equation.
Method of Elimination: A method for solving systems of linear equations by adding or subtracting the equations to eliminate one of the variables.
Linear Programming: A method for solving optimization problems subject to certain constraints.
Computer Graphics: The use of computers to create and manipulate visual images.
Physics: The study of the natural world around us.

References

"Linear Algebra and Its Applications" by Gilbert Strang
"Introduction to Linear Algebra" by Gilbert Strang
"Linear Algebra and Its Applications" by David C. Lay