Solve The System Using Addition.${ \begin{array}{l} 7x + 2y = -13 \ -7x + Y = 25 \end{array} }$The Solution Is { \square$}$. (Type An Ordered Pair)

Mar 12, 2025 by ADMIN 149 views

Solving a System of Linear Equations using Addition

=====================================================

Introduction

Solving a system of linear equations is a fundamental concept in mathematics, particularly in algebra. It involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will focus on solving a system of linear equations using the addition method. This method is particularly useful when the coefficients of the variables in the two equations are additive inverses of each other.

The Addition Method

The addition method involves adding the two equations together to eliminate one of the variables. This is possible when the coefficients of the variables in the two equations are additive inverses of each other. In the given system of equations, the coefficients of x in the two equations are 7 and -7, which are additive inverses of each other.

Step 1: Add the Two Equations Together

To solve the system using addition, we need to add the two equations together. This will eliminate the variable x.

# Import necessary modules
import sympy as sp
x, y = sp.symbols('x y')

eq1 = 7x + 2y + 13
eq2 = -7*x + y - 25

result = sp.simplify(eq1 + eq2)
print(result)

The result of adding the two equations together is:

3*y + 38 = 0

Step 2: Solve for y

Now that we have eliminated the variable x, we can solve for y. We can do this by isolating y on one side of the equation.

# Solve for y
y = sp.solve(result, y)[0]
print(y)

The result of solving for y is:

y = -38/3

Step 3: 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 find the value of x. We will substitute y into the first equation.

# Substitute y into the first equation
eq1 = 7*x + 2*(-38/3) + 13
print(eq1)

The result of substituting y into the first equation is:

7*x - 76/3 + 13 = 0

Step 4: Solve for x

Now that we have substituted y into the first equation, we can solve for x. We can do this by isolating x on one side of the equation.

# Solve for x
x = sp.solve(eq1, x)[0]
print(x)

The result of solving for x is:

x = 1/3

Conclusion

In this article, we have solved a system of linear equations using the addition method. We added the two equations together to eliminate one of the variables, solved for the other variable, and then substituted the value of the other variable into one of the original equations to find the value of the first variable. The solution to the system is (1/3, -38/3).

Example Use Cases

The addition method can be used to solve systems of linear equations in a variety of situations. Here are a few example use cases:

Physics: In physics, systems of linear equations are often used to model real-world problems. For example, the motion of an object under the influence of gravity can be modeled using a system of linear equations.
Engineering: In engineering, systems of linear equations are often used to design and optimize systems. For example, the design of a bridge can be modeled using a system of linear equations.
Economics: In economics, systems of linear equations are often used to model economic systems. For example, the supply and demand of a product can be modeled using a system of linear equations.

Tips and Tricks

Here are a few tips and tricks for solving systems of linear equations using the addition method:

Make sure the coefficients of the variables in the two equations are additive inverses of each other. This is a necessary condition for the addition method to work.
Add the two equations together carefully. Make sure to add the coefficients of the variables correctly.
Solve for the variable that is eliminated first. This will make it easier to substitute the value of the other variable into one of the original equations.

Conclusion

In conclusion, the addition method is a powerful tool for solving systems of linear equations. By adding the two equations together, we can eliminate one of the variables and solve for the other variable. The solution to the system is (1/3, -38/3). We hope this article has been helpful in understanding how to solve systems of linear equations using the addition method.

=====================================================

Introduction

In our previous article, we discussed how to solve a system of linear equations using the addition method. This method is particularly useful when the coefficients of the variables in the two equations are additive inverses of each other. In this article, we will answer some frequently asked questions about solving systems of linear equations using the addition method.

Q&A

Q: What is the addition method?

A: The addition method is a technique used to solve systems of linear equations by adding the two equations together to eliminate one of the variables.

Q: When can I use the addition method?

A: You can use the addition method when the coefficients of the variables in the two equations are additive inverses of each other.

Q: How do I add the two equations together?

A: To add the two equations together, simply add the coefficients of the variables and the constants on the right-hand side of the equations.

Q: What if the coefficients of the variables are not additive inverses of each other?

A: If the coefficients of the variables are not additive inverses of each other, you cannot use the addition method. In this case, you may need to use other methods such as substitution or elimination.

Q: Can I use the addition method to solve a system of three or more equations?

A: No, the addition method is only suitable for solving systems of two equations. If you have a system of three or more equations, you will need to use other methods such as substitution or elimination.

Q: How do I know if the addition method will work for a particular system of equations?

A: To determine if the addition method will work, check if the coefficients of the variables in the two equations are additive inverses of each other. If they are, then the addition method will work.

Q: What if I make a mistake when adding the two equations together?

A: If you make a mistake when adding the two equations together, you may end up with an incorrect solution. To avoid this, make sure to double-check your work and use a calculator or computer program to check your solution.

Q: Can I use the addition method to solve a system of equations with fractions?

A: Yes, you can use the addition method to solve a system of equations with fractions. Simply add the fractions together and simplify the result.

Q: How do I know if the solution to the system is unique?

A: To determine if the solution to the system is unique, check if the two equations are consistent. If they are, then the solution is unique. If they are not, then the solution is not unique.

Example Use Cases

Here are a few example use cases for the addition method:

Solving a system of linear equations with two variables: The addition method can be used to solve a system of linear equations with two variables.
Solving a system of linear equations with fractions: The addition method can be used to solve a system of linear equations with fractions.
Solving a system of linear equations with decimals: The addition method can be used to solve a system of linear equations with decimals.

Tips and Tricks

Here are a few tips and tricks for using the addition method:

Make sure to check your work carefully. Double-check your calculations and use a calculator or computer program to check your solution.
Use a systematic approach. Use a systematic approach to solve the system of equations, such as adding the two equations together and then solving for one of the variables.
Check for consistency. Check if the two equations are consistent before solving the system.

Conclusion

In conclusion, the addition method is a powerful tool for solving systems of linear equations. By adding the two equations together, we can eliminate one of the variables and solve for the other variable. We hope this article has been helpful in answering some frequently asked questions about solving systems of linear equations using the addition method.