Understanding the Method of Elimination

Oct 13, 2025 by windailys.com 53 views

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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 using the method of substitution and elimination. We will use the given equations:

{ \begin{array}{l} 5x + 6y = 75 \\ 5x + 3y = 60 \end{array} \}

to find the solution $(x, y)$ .

Understanding the Method of Elimination

The method of elimination is a technique used to solve a system of linear equations by adding or subtracting the equations to eliminate one of the variables. In this case, we will subtract the second equation from the first equation to eliminate the variable $x$ .

Subtracting the Equations

To subtract the equations, we need to make the coefficients of $x$ the same in both equations. Since the coefficients of $x$ are already the same (5), we can simply subtract the second equation from the first equation:

{ \begin{array}{l} (5x + 6y) - (5x + 3y) = 75 - 60 \\ 3y = 15 \end{array} \}

Solving for $y$

Now that we have eliminated the variable $x$ , we can solve for $y$ by dividing both sides of the equation by 3:

{ \begin{array}{l} 3y = 15 \\ y = \frac{15}{3} \\ y = 5 \end{array} \}

Substituting $y$ into One of the Original Equations

Now that we have found the value of $y$ , we can substitute it into one of the original equations to find the value of $x$ . We will use the first equation:

{ \begin{array}{l} 5x + 6y = 75 \\ 5x + 6(5) = 75 \\ 5x + 30 = 75 \end{array} \}

Solving for $x$

Now that we have substituted the value of $y$ into the equation, we can solve for $x$ by subtracting 30 from both sides of the equation and then dividing both sides by 5:

{ \begin{array}{l} 5x + 30 = 75 \\ 5x = 45 \\ x = \frac{45}{5} \\ x = 9 \end{array} \}

Conclusion

In this article, we have used the method of elimination to solve a system of two linear equations. We have subtracted the equations to eliminate the variable $x$ and then solved for $y$ . Finally, we have substituted the value of $y$ into one of the original equations to find the value of $x$ . The solution to the system of linear equations is $(x, y) = (9, 5)$ .

Example Use Cases

The method of elimination can be used to solve a wide range of systems of linear equations. Some example use cases include:

Physics: When solving problems involving motion, forces, and energies, systems of linear equations are often used to model the relationships between different physical quantities.
Engineering: In fields such as civil engineering, mechanical engineering, and electrical engineering, systems of linear equations are used to design and analyze complex systems.
Economics: In economics, systems of linear equations are used to model the relationships between different economic variables, such as supply and demand.

Tips and Tricks

When using the method of elimination to solve a system of linear equations, here are a few tips and tricks to keep in mind:

Make sure to subtract the equations correctly: When subtracting the equations, make sure to subtract the corresponding terms.
Check your work: After solving for the variables, check your work by plugging the values back into the original equations.
Use the method of substitution as an alternative: If the method of elimination is not working, try using the method of substitution as an alternative.

Conclusion

In conclusion, the method of elimination is a powerful tool for solving systems of linear equations. By subtracting the equations and solving for the variables, we can find the solution to the system. With practice and patience, you can master the method of elimination and become proficient in solving systems of linear equations.

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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 different methods for solving systems of linear equations?

A: There are two main methods for solving systems of linear equations: the method of substitution and the method of elimination. The method of substitution involves solving one equation 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.

Q: What is the method of elimination?

A: The method of elimination is a technique used to solve a system of linear equations by adding or subtracting the equations to eliminate one of the variables.

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

A: To use the method of elimination, follow these steps:

Make the coefficients of one of the variables the same in both equations.
Add or subtract the equations to eliminate one of the variables.
Solve for the remaining variable.
Substitute the value of the remaining variable into one of the original equations to find the value of the other variable.

Q: What are some common mistakes to avoid when using the method of elimination?

A: Some common mistakes to avoid when using the method of elimination include:

Subtracting the equations incorrectly
Not checking your work
Not using the correct method for the given system of equations

Q: Can I use the method of substitution to solve a system of linear equations?

A: Yes, you can use the method of substitution to solve a system of linear equations. This method involves solving one equation for one variable and then substituting that expression into the other equation.

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

A: Solving systems of linear equations has many real-world applications, including:

Physics: Solving systems of linear equations is used to model the relationships between different physical quantities, such as motion, forces, and energies.
Engineering: Solving systems of linear equations is used to design and analyze complex systems, such as bridges, buildings, and electronic circuits.
Economics: Solving systems of linear equations is used to model the relationships between different economic variables, such as supply and demand.

Q: How do I know which method to use when solving a system of linear equations?

A: To determine which method to use, follow these steps:

Look at the coefficients of the variables in the equations.
If the coefficients are the same, use the method of elimination.
If the coefficients are different, use the method of substitution.

Q: Can I use a calculator to solve a system of linear equations?

A: Yes, you can use a calculator to solve a system of linear equations. Many calculators have built-in functions for solving systems of linear equations, such as the "solve" function.

Q: What are some tips for solving systems of linear equations?

A: Some tips for solving systems of linear equations include:

Make sure to follow the steps carefully.
Check your work to ensure that the solution is correct.
Use the correct method for the given system of equations.
Practice, practice, practice!

Conclusion

In conclusion, solving systems of linear equations is an important skill that has many real-world applications. By understanding the different methods for solving systems of linear equations, you can become proficient in solving these types of problems. Remember to practice regularly and to check your work to ensure that the solution is correct.