Not verifying the solution

Oct 13, 2025 by pynkocean.com 21 views

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Introduction

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of two linear equations with two variables. We will use the given system of equations as an example to demonstrate the steps involved in solving it.

The System of Equations

The given system of equations is:

\begin{array}{l} y = -2 \\ 4x - 3y = 18 \end{array}

Step 1: Identify the Equations

The first equation is a simple linear equation in one variable, y. The second equation is a linear equation in two variables, x and y.

Step 2: Substitute the Value of y

Since the value of y is already given in the first equation, we can substitute it into the second equation to eliminate the variable y.

4x - 3(-2) = 18

Step 3: Simplify the Equation

Simplifying the equation, we get:

4x + 6 = 18

Step 4: Isolate the Variable x

Subtracting 6 from both sides of the equation, we get:

4x = 12

Step 5: Solve for x

Dividing both sides of the equation by 4, we get:

x = 3

Step 6: Verify the Solution

Now that we have found the value of x, we can substitute it back into one of the original equations to verify the solution. Let's use the first equation:

y = -2

Substituting x = 3 into the second equation, we get:

4(3) - 3(-2) = 18

Simplifying the equation, we get:

12 + 6 = 18

Which is true. Therefore, the solution is verified.

Conclusion

In this article, we have demonstrated the steps involved in solving a system of linear equations with two variables. We used the given system of equations as an example to illustrate the process. By following these steps, we were able to find the values of x and y that satisfy both equations simultaneously.

Tips and Tricks

When solving a system of linear equations, it is essential to identify the equations and the variables involved.
Substituting the value of one variable into the other equation can help eliminate the variable and simplify the equation.
Verifying the solution by substituting the values back into the original equations is crucial to ensure that the solution is correct.

Real-World Applications

Solving systems of linear equations has numerous real-world applications in various fields, including:

Physics: Solving systems of linear equations is essential in physics to describe the motion of objects and the forces acting upon them.
Engineering: Systems of linear equations are used in engineering to design and analyze complex systems, such as electrical circuits and mechanical systems.
Economics: Solving systems of linear equations is used in economics to model and analyze economic systems, including supply and demand curves.

Final Thoughts

Solving systems of linear equations is a fundamental concept in mathematics that has numerous real-world applications. By following the steps outlined in this article, you can solve systems of linear equations with ease and confidence. Remember to identify the equations and variables involved, substitute the value of one variable into the other equation, and verify the solution by substituting the values back into the original equations.

Additional Resources

For further learning and practice, here are some additional resources:

Khan Academy: Systems of Linear Equations
MIT OpenCourseWare: Linear Algebra
Wolfram Alpha: Systems of Linear Equations

Frequently Asked Questions

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 two or more variables.
Q: How do I solve a system of linear equations?
A: To solve a system of linear equations, you can use the steps outlined in this article, including identifying the equations and variables, substituting the value of one variable into the other equation, and verifying the solution.
Q: What are some real-world applications of solving systems of linear equations?
A: Solving systems of linear equations has numerous real-world applications in physics, engineering, economics, and other fields.

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Introduction

Solving systems of linear equations is a fundamental concept in mathematics that has numerous real-world applications. However, it can be a challenging topic for many students and professionals. In this article, we will address some of the most frequently asked questions about 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 involve two or more variables. Each equation is a linear equation, meaning that it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

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

A: To solve a system of linear equations, you can use the following steps:

Identify the equations and variables involved.
Determine the method of solution, such as substitution or elimination.
Use the chosen method to solve for one variable.
Substitute the value of the solved variable back into one of the original equations to solve for the other variable.
Verify the solution by substituting the values back into the original equations.

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

A: There are two main methods of solving systems of linear equations:

Substitution Method: This method involves substituting the value of one variable into the other equation to solve for the other variable.
Elimination Method: This method involves adding or subtracting the equations to eliminate one of the variables.

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

A: The substitution method involves substituting the value of one variable into the other equation, while the elimination method involves adding or subtracting the equations to eliminate one of the variables.

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

A: The choice between the substitution and elimination methods depends on the specific system of equations. If one of the equations is already solved for one variable, the substitution method may be easier to use. If the coefficients of the variables are the same, the elimination method may be easier to use.

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 verifying the solution: Failing to verify the solution by substituting the values back into the original equations can lead to incorrect answers.
Not following the correct order of operations: Failing to follow the correct order of operations, such as PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction), can lead to incorrect answers.
Not checking for extraneous solutions: Failing to check for extraneous solutions can lead to incorrect answers.

Q: What are some real-world applications of solving systems of linear equations?

A: Solving systems of linear equations has numerous real-world applications in physics, engineering, economics, and other fields. Some examples include:

Physics: Solving systems of linear equations is essential in physics to describe the motion of objects and the forces acting upon them.
Engineering: Systems of linear equations are used in engineering to design and analyze complex systems, such as electrical circuits and mechanical systems.
Economics: Solving systems of linear equations is used in economics to model and analyze economic systems, including supply and demand curves.

Q: How can I practice solving systems of linear equations?

A: There are many resources available to practice solving systems of linear equations, including:

Online resources: Websites such as Khan Academy, MIT OpenCourseWare, and Wolfram Alpha offer interactive lessons and practice problems.
Textbooks: Many algebra and mathematics textbooks include practice problems and exercises to help you practice solving systems of linear equations.
Practice tests: Taking practice tests can help you assess your understanding of solving systems of linear equations and identify areas for improvement.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics that has numerous real-world applications. By understanding the different methods of solution, avoiding common mistakes, and practicing with real-world examples, you can become proficient in solving systems of linear equations. Remember to verify your solutions, follow the correct order of operations, and check for extraneous solutions to ensure that your answers are accurate.