Why We Chose the Substitution Method

Oct 13, 2025 by pynkocean.com 10 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 given system of equations as an example and explain the method we chose to solve it.

The System of Equations

The system of equations we will be solving is:

\begin{align*} x &= 3y + 7 \\ 2x + 12y &= -4 \end{align*}

Choosing a Method

There are several methods to solve a system of linear equations, including substitution, elimination, and graphing. In this case, we will use the substitution method.

The Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. In this case, we will solve the first equation for x and then substitute that expression into the second equation.

Step 1: Solve the First Equation for x

We will solve the first equation for x:

x = 3y + 7

Step 2: Substitute the Expression into the Second Equation

We will substitute the expression for x into the second equation:

2(3y + 7) + 12y = -4

Step 3: Simplify the Equation

We will simplify the equation by combining like terms:

6y + 14 + 12y = -4

Step 4: Combine Like Terms

We will combine like terms:

18y + 14 = -4

Step 5: Subtract 14 from Both Sides

We will subtract 14 from both sides:

18y = -18

Step 6: Divide Both Sides by 18

We will divide both sides by 18:

y = -1

Step 7: Substitute the Value of y into the First Equation

We will substitute the value of y into the first equation:

x = 3(-1) + 7

Step 8: Simplify the Equation

We will simplify the equation:

x = -3 + 7

Step 9: Combine Like Terms

We will combine like terms:

x = 4

Conclusion

In this article, we solved a system of two linear equations with two variables using the substitution method. We chose this method because it is a straightforward and easy-to-understand method for solving systems of linear equations. We solved the first equation for x and then substituted that expression into the second equation. We simplified the equation and combined like terms to find the value of y. Finally, we substituted the value of y into the first equation to find the value of x.

Answer

The solution to the system of equations is:

x = 4

y = -1

Why We Chose the Substitution Method

We chose the substitution method because it is a simple and intuitive method for solving systems of linear equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This method is easy to understand and is a good choice for solving systems of linear equations with two variables.

Advantages of the Substitution Method

The substitution method has several advantages, including:

It is a simple and intuitive method for solving systems of linear equations.
It is easy to understand and apply.
It is a good choice for solving systems of linear equations with two variables.

Disadvantages of the Substitution Method

The substitution method has several disadvantages, including:

It may not be the best choice for solving systems of linear equations with more than two variables.
It may require more steps than other methods, such as the elimination method.

Conclusion

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 several methods for solving systems of linear equations, including:

Substitution method
Elimination method
Graphing method
Matrix method

Q: What is the substitution method?

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

Q: What is the elimination method?

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

Q: What is the graphing method?

A: The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the matrix method?

A: The matrix method involves using matrices to solve the system of equations.

Q: Which method is the best to use?

A: The best method to use depends on the specific system of equations and the variables involved. The substitution method is often the easiest to use, but the elimination method may be more efficient for some systems.

Q: How do I know which method to use?

A: You can try using the substitution method first, and if it doesn't work, try using the elimination method. If neither method works, you can try using the graphing method or the matrix method.

Q: What if I have a system of linear equations with more than two variables?

A: If you have a system of linear equations with more than two variables, you may need to use a more advanced method, such as the matrix method or the elimination method.

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.

Q: How do I check my answer?

A: To check your answer, you can plug the values of the variables back into the original equations and make sure that they are true.

Q: What if I make a mistake while solving a system of linear equations?

A: If you make a mistake while solving a system of linear equations, you can try reworking the problem or using a different method to solve it.

Conclusion

In conclusion, solving systems of linear equations can be a challenging task, but with the right methods and techniques, it can be done. The substitution method, elimination method, graphing method, and matrix method are all useful tools for solving systems of linear equations. By understanding the different methods and choosing the right one for the problem, you can solve systems of linear equations with ease.

Additional Resources

Online calculators for solving systems of linear equations
Math textbooks and workbooks for additional practice and review
Online resources and tutorials for learning more about solving systems of linear equations

Final Tips

Practice, practice, practice! The more you practice solving systems of linear equations, the more comfortable you will become with the different methods and techniques.
Don't be afraid to ask for help if you need it. Whether it's a teacher, a tutor, or a classmate, there are many resources available to help you with solving systems of linear equations.
Take your time and be patient. Solving systems of linear equations can be a challenging task, but with persistence and practice, you can master it.