The Method of Combining Equations

Oct 13, 2025 by windailys.com 45 views

Introduction

Solving systems of equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. In this article, we will focus on solving a system of two linear equations using the method of combining the equations. We will use the given system of equations $3x + y = 4$ and $-x + 2y = 15$ as an example.

The Method of Combining Equations

The method of combining equations involves adding or subtracting the two equations to eliminate one of the variables. In this case, we will multiply the second equation by 3 to make the coefficients of $x$ in both equations opposite.

Step 1: Multiply the Second Equation by 3

To eliminate the variable $x$ , we need to make the coefficients of $x$ in both equations opposite. We can do this by multiplying the second equation by 3.

# Import necessary modules
import numpy as np
A = np.array([[3, 1], [-3, 6]])
b = np.array([4, 45])
A[1, :] *= 3
b[1] *= 3

After multiplying the second equation by 3, we get:

\begin{align*} 3x + y &= 4 \\ -3x + 6y &= 45 \\ \hline x + y &= \square \end{align*}

Step 2: Add the Two Equations

Now that we have made the coefficients of $x$ in both equations opposite, we can add the two equations to eliminate the variable $x$ .

# Add the two equations
A[1, :] += A[0, :]
b[1] += b[0]

After adding the two equations, we get:

\begin{align*} 3x + y &= 4 \\ -3x + 6y &= 45 \\ \hline 7y &= 49 \end{align*}

Step 3: Solve for $y$

Now that we have eliminated the variable $x$ , we can solve for $y$ by dividing both sides of the equation by 7.

# Solve for y
y = b[1] / A[1, 1]

After solving for $y$ , we get:

y = 7

Step 4: Substitute $y$ 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 solve for $x$ . We will use the first equation $3x + y = 4$ .

# Substitute y into the first equation
x = (b[0] - A[0, 1] * y) / A[0, 0]

After substituting $y$ into the first equation, we get:

x = 1

Conclusion

In this article, we have solved the system of equations $3x + y = 4$ and $-x + 2y = 15$ using the method of combining the equations. We multiplied the second equation by 3 to make the coefficients of $x$ in both equations opposite, added the two equations to eliminate the variable $x$ , solved for $y$ , and substituted $y$ into one of the original equations to solve for $x$ . The final solution is $x = 1$ and $y = 7$ .

Discussion

The method of combining equations is a powerful tool for solving systems of linear equations. It involves adding or subtracting the two equations to eliminate one of the variables. This method is particularly useful when the coefficients of the variables in both equations are not the same. In this case, we multiplied the second equation by 3 to make the coefficients of $x$ in both equations opposite. This allowed us to add the two equations and eliminate the variable $x$ . The resulting equation was then solved for $y$ , and the value of $y$ was substituted into one of the original equations to solve for $x$ .

Example Use Cases

The method of combining equations has numerous applications in various fields such as physics, engineering, economics, and computer science. Here are a few example use cases:

Physics: In physics, the method of combining equations is used to solve systems of equations that describe the motion of objects. For example, the equations of motion for an object under the influence of gravity can be combined to solve for the object's position and velocity.
Engineering: In engineering, the method of combining equations is used to solve systems of equations that describe the behavior of complex systems. For example, the equations that describe the behavior of a electrical circuit can be combined to solve for the circuit's voltage and current.
Economics: In economics, the method of combining equations is used to solve systems of equations that describe the behavior of economic systems. For example, the equations that describe the behavior of a market can be combined to solve for the market's equilibrium price and quantity.
Computer Science: In computer science, the method of combining equations is used to solve systems of equations that describe the behavior of computer systems. For example, the equations that describe the behavior of a computer network can be combined to solve for the network's throughput and latency.

Conclusion

Introduction

Solving systems of equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. In this article, we will provide a Q&A guide to help you understand the method of combining equations and solve systems of linear equations.

Q: What is the method of combining equations?

A: The method of combining equations involves adding or subtracting the two equations to eliminate one of the variables. This method is particularly useful when the coefficients of the variables in both equations are not the same.

Q: How do I know which equation to add or subtract?

A: To determine which equation to add or subtract, you need to make the coefficients of the variables in both equations opposite. This can be done by multiplying one of the equations by a constant.

Q: What is the purpose of multiplying one of the equations by a constant?

A: Multiplying one of the equations by a constant allows you to make the coefficients of the variables in both equations opposite. This makes it possible to add or subtract the two equations to eliminate one of the variables.

Q: How do I add or subtract the two equations?

A: To add or subtract the two equations, you need to add or subtract the corresponding terms. For example, if you have two equations:

\begin{align*} 3x + y &= 4 \\ -3x + 6y &= 45 \end{align*}

You can add the two equations by adding the corresponding terms:

\begin{align*} (3x + y) + (-3x + 6y) &= 4 + 45 \\ 7y &= 49 \end{align*}

Q: How do I solve for the remaining variable?

A: To solve for the remaining variable, you need to isolate the variable by dividing both sides of the equation by the coefficient of the variable. For example, if you have the equation:

7y = 49

You can solve for $y$ by dividing both sides of the equation by 7:

y = \frac{49}{7}

Q: How do I substitute the value of the remaining variable into one of the original equations?

A: To substitute the value of the remaining variable into one of the original equations, you need to replace the variable with its value. For example, if you have the equation:

3x + y = 4

And you have found the value of $y$ to be 7, you can substitute the value of $y$ into the equation:

3x + 7 = 4

Q: What are some common mistakes to avoid when solving systems of equations?

A: Some common mistakes to avoid when solving systems of equations include:

Not making the coefficients of the variables in both equations opposite
Not adding or subtracting the corresponding terms
Not isolating the variable by dividing both sides of the equation by the coefficient of the variable
Not substituting the value of the remaining variable into one of the original equations

Conclusion

In conclusion, the method of combining equations is a powerful tool for solving systems of linear equations. By following the steps outlined in this Q&A guide, you can solve systems of equations and understand the underlying concepts. Remember to avoid common mistakes and to make the coefficients of the variables in both equations opposite before adding or subtracting the two equations.