Given the System of Equations

Oct 13, 2025 by windailys.com 2 views

Introduction

In mathematics, solving systems of equations is a fundamental concept that involves finding the values of variables that satisfy multiple equations simultaneously. One of the methods used to solve systems of equations is the method of replacement, which involves replacing one equation with the sum of that equation and a multiple of the other equation. In this article, we will show and explain how replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions as the one shown.

The Method of Replacement

The method of replacement is a technique used to solve systems of equations by replacing one equation with the sum of that equation and a multiple of the other equation. This method is based on the concept of linear combinations, which states that any linear combination of two equations is also an equation in the system.

Given the System of Equations

Let's consider the following system of equations:

{ \begin{align*} 8x + 7y &= 39 \\ 4x - 14y &= -26 \end{align*} }

Our goal is to show that replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions as the one shown.

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

To replace one equation by the sum of that equation and a multiple of the other, we need to find a multiple of the second equation that, when added to the first equation, will eliminate one of the variables. Let's try to eliminate the variable $x$ .

We can multiply the second equation by $2$ to get:

{ \begin{align*} 8x + 7y &= 39 \\ 8x - 28y &= -52 \end{align*} }

Now, we can add the two equations to eliminate the variable $x$ :

{ \begin{align*} (8x + 7y) + (8x - 28y) &= 39 + (-52) \\ 16x - 21y &= -13 \end{align*} }

However, we want to eliminate the variable $x$ , not $y$ . To do this, we need to multiply the second equation by $-2$ instead of $2$ . This will give us:

{ \begin{align*} 8x + 7y &= 39 \\ -8x + 28y &= 52 \end{align*} }

Now, we can add the two equations to eliminate the variable $x$ :

{ \begin{align*} (8x + 7y) + (-8x + 28y) &= 39 + 52 \\ 35y &= 91 \end{align*} }

This is the new system of equations that we obtained by replacing one equation by the sum of that equation and a multiple of the other.

Solving the New System of Equations

Now that we have the new system of equations, we can solve for the variable $y$ . We can divide both sides of the equation by $35$ to get:

{ \begin{align*} y &= \frac{91}{35} \\ y &= \frac{13}{5} \end{align*} }

Now that we have the value of $y$ , we can substitute it back into one of the original equations to solve for the variable $x$ . Let's use the first equation:

{ \begin{align*} 8x + 7y &= 39 \\ 8x + 7\left(\frac{13}{5}\right) &= 39 \\ 8x + \frac{91}{5} &= 39 \\ 8x &= 39 - \frac{91}{5} \\ 8x &= \frac{195 - 91}{5} \\ 8x &= \frac{104}{5} \\ x &= \frac{104}{5 \cdot 8} \\ x &= \frac{13}{5} \end{align*} }

Therefore, the solution to the system of equations is $x = \frac{13}{5}$ and $y = \frac{13}{5}$ .

Conclusion

In this article, we showed and explained how replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions as the one shown. We used the method of replacement to solve a system of equations and obtained the same solution as the original system. This method is a powerful tool for solving systems of equations and can be used to solve a wide range of problems in mathematics and other fields.

References

[1] "Solving Systems of Equations" by Math Open Reference
[2] "Method of Replacement" by Khan Academy
[3] "Linear Combinations" by Wolfram MathWorld

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are related to each other through the variables in the equations. In other words, a system of equations is a collection of equations that must be solved simultaneously to find the values of the variables.

Q: What is the method of replacement?

A: The method of replacement is a technique used to solve systems of equations by replacing one equation with the sum of that equation and a multiple of the other equation. This method is based on the concept of linear combinations, which states that any linear combination of two equations is also an equation in the system.

Q: How do I know which equation to replace?

A: To determine which equation to replace, you need to look at the coefficients of the variables in the two equations. If the coefficients of one variable are the same in both equations, you can replace one equation with the sum of that equation and a multiple of the other equation.

Q: What is a linear combination?

A: A linear combination is an equation that is formed by adding or subtracting a multiple of one equation from another equation. In other words, a linear combination is an equation that is a combination of two or more equations.

Q: How do I solve a system of equations using the method of replacement?

A: To solve a system of equations using the method of replacement, follow these steps:

Identify the two equations in the system.
Determine which equation to replace by looking at the coefficients of the variables.
Replace one equation with the sum of that equation and a multiple of the other equation.
Solve the new equation for one of the variables.
Substitute the value of the variable back into one of the original equations to solve for the other variable.

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

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

Not checking the coefficients of the variables before replacing one equation with the sum of that equation and a multiple of the other equation.
Not solving the new equation for one of the variables.
Not substituting the value of the variable back into one of the original equations to solve for the other variable.

Q: Can the method of replacement be used to solve systems of equations with more than two equations?

A: Yes, the method of replacement can be used to solve systems of equations with more than two equations. However, it may be more difficult to determine which equation to replace and which multiple to use.

Q: Are there any other methods for solving systems of equations?

A: Yes, there are several other methods for solving systems of equations, including:

The method of substitution
The method of elimination
The method of graphing
The method of matrices

Each of these methods has its own strengths and weaknesses, and the choice of method will depend on the specific system of equations being solved.

Q: How do I know which method to use?

A: The choice of method will depend on the specific system of equations being solved. If the system has two equations and two variables, the method of replacement or the method of substitution may be the best choice. If the system has more than two equations or variables, the method of elimination or the method of matrices may be more suitable.

Conclusion

In this article, we have answered some of the most frequently asked questions about solving systems of equations using the method of replacement. We have discussed the concept of linear combinations, the steps involved in solving a system of equations using the method of replacement, and some common mistakes to avoid. We have also discussed other methods for solving systems of equations and how to choose the best method for a given system.