Replacing One Equation with the Sum of That Equation and a Multiple of the Other

Oct 13, 2025 by magicalceylontours.com 100 views

Introduction

In mathematics, solving systems of linear equations is a fundamental concept that has numerous applications in various fields, including physics, engineering, and economics. A system of linear equations consists of two or more linear equations that involve the same set of variables. In this article, we will explore a technique for solving systems of linear equations by replacing one equation with the sum of that equation and a multiple of the other.

The Given System

The given system of linear equations is:

\begin{align*} 1. & \quad 8x + 7y = 39 \\ 2. & \quad 4x - 14y = 13 \end{align*}

Replacing One Equation with the Sum of That Equation and a Multiple of the Other

To solve this system, we will replace equation (1) with the sum of equation (1) and a multiple of equation (2). Let's choose a multiple of equation (2) that will eliminate the variable $y$ when added to equation (1). We can multiply equation (2) by $\frac{7}{14} = \frac{1}{2}$ to get:

\begin{align*} 2. & \quad 2x - 7y = \frac{13}{2} \end{align*}

Now, we will add equation (1) and the modified equation (2) to get:

\begin{align*} (8x + 7y) + (2x - 7y) & = 39 + \frac{13}{2} \\ 10x & = 39 + \frac{13}{2} \\ 10x & = \frac{87}{2} \\ x & = \frac{87}{20} \end{align*}

Substituting the Value of $x$ into One of the Original Equations

Now that we have found the value of $x$ , we can substitute it into one of the original equations to find the value of $y$ . Let's substitute $x = \frac{87}{20}$ into equation (1):

\begin{align*} 8\left(\frac{87}{20}\right) + 7y & = 39 \\ \frac{87}{5} + 7y & = 39 \\ 7y & = 39 - \frac{87}{5} \\ 7y & = \frac{195 - 87}{5} \\ 7y & = \frac{108}{5} \\ y & = \frac{108}{35} \end{align*}

Conclusion

In this article, we have shown how replacing one equation with the sum of that equation and a multiple of the other produces a system with the same solutions as the original system. We have applied this technique to the given system of linear equations and found the values of $x$ and $y$ . This technique is a powerful tool for solving systems of linear equations and has numerous applications in various fields.

Step-by-Step Solution

Here is a step-by-step solution to the given system of linear equations:

Replace equation (1) with the sum of equation (1) and a multiple of equation (2):

\begin{align*} (8x + 7y) + (2x - 7y) & = 39 + \frac{13}{2} \\ 10x & = 39 + \frac{13}{2} \\ 10x & = \frac{87}{2} \\ x & = \frac{87}{20} \end{align*}

Substitute the value of $x$ into one of the original equations:

\begin{align*} 8\left(\frac{87}{20}\right) + 7y & = 39 \\ \frac{87}{5} + 7y & = 39 \\ 7y & = 39 - \frac{87}{5} \\ 7y & = \frac{195 - 87}{5} \\ 7y & = \frac{108}{5} \\ y & = \frac{108}{35} \end{align*}

Final Answer

Q: What is the purpose of replacing one equation with the sum of that equation and a multiple of the other?

A: The purpose of replacing one equation with the sum of that equation and a multiple of the other is to eliminate one of the variables in the system of linear equations. This technique is useful when the system has two equations with two variables, and we want to find the values of the variables.

Q: How do I choose the multiple of the other equation?

A: To choose the multiple of the other equation, we need to find a multiple that will eliminate the variable $y$ when added to equation (1). We can multiply equation (2) by $\frac{7}{14} = \frac{1}{2}$ to get:

\begin{align*} 2. & \quad 2x - 7y = \frac{13}{2} \end{align*}

Q: What if I choose a different multiple of the other equation? Will it affect the solution?

A: Yes, choosing a different multiple of the other equation will affect the solution. If we choose a multiple that does not eliminate the variable $y$ , we will not be able to find the values of the variables.

Q: Can I use this technique to solve systems of linear equations with more than two variables?

A: No, this technique is only applicable to systems of linear equations with two variables. If we have a system with more than two variables, we will need to use a different technique, such as substitution or elimination.

Q: What if I have a system of linear equations with no solution? Can I still use this technique?

A: Yes, you can still use this technique even if the system has no solution. However, if the system has no solution, the values of the variables will be undefined.

Q: How do I know if the system has a unique solution, infinitely many solutions, or no solution?

A: To determine if the system has a unique solution, infinitely many solutions, or no solution, we need to check the rank of the augmented matrix. If the rank of the augmented matrix is equal to the number of variables, the system has a unique solution. If the rank of the augmented matrix is less than the number of variables, the system has infinitely many solutions. If the rank of the augmented matrix is greater than the number of variables, the system has no solution.

Q: Can I use this technique to solve systems of linear equations with complex coefficients?

A: Yes, you can use this technique to solve systems of linear equations with complex coefficients. However, you will need to use complex arithmetic to perform the calculations.

Q: What are some common mistakes to avoid when using this technique?

A: Some common mistakes to avoid when using this technique include:

Choosing a multiple of the other equation that does not eliminate the variable $y$
Not checking the rank of the augmented matrix to determine if the system has a unique solution, infinitely many solutions, or no solution
Not performing the calculations correctly, such as multiplying the equations by the wrong multiples or adding the equations incorrectly.

Conclusion

In this article, we have answered some frequently asked questions about replacing one equation with the sum of that equation and a multiple of the other to solve systems of linear equations. We have discussed the purpose of this technique, how to choose the multiple of the other equation, and some common mistakes to avoid. We hope that this article has been helpful in understanding this technique and how to use it to solve systems of linear equations.