Solve For A A A :${ 6a + 5a = -11 }$

Apr 2, 2025 by ADMIN 38 views

**Solving Linear Equations: A Step-by-Step Guide**

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific type of linear equation, where the variable is isolated on one side of the equation. We will use the given equation $6a + 5a = -11$ as an example to demonstrate the step-by-step process of solving for $a$ .

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at what it means. The equation $6a + 5a = -11$ is a linear equation, which means it is an equation in which the highest power of the variable (in this case, $a$ ) is 1. The equation is also a simple linear equation, which means it can be solved using basic algebraic operations.

Step 1: Combine Like Terms

The first step in solving the equation is to combine like terms. In this case, we have two terms with the variable $a$ , which are $6a$ and $5a$ . We can combine these terms by adding their coefficients (the numbers in front of the variable). So, $6a + 5a$ can be simplified to $11a$ .

# Combine like terms
a = sympy.Symbol('a')
equation = 6*a + 5*a
simplified_equation = sympy.simplify(equation)
print(simplified_equation)  # Output: 11*a

Step 2: Isolate the Variable

Now that we have simplified the equation to $11a = -11$ , we need to isolate the variable $a$ . To do this, we can divide both sides of the equation by 11. This will give us the value of $a$ .

# Isolate the variable
equation = 11*a + 11
solution = sympy.solve(equation, a)
print(solution)  # Output: [-1]

Step 3: Check the Solution

Once we have isolated the variable, we need to check our solution to make sure it is correct. We can do this by plugging the value of $a$ back into the original equation and checking if it is true.

# Check the solution
a = -1
equation = 6*a + 5*a + 11
print(equation)  # Output: 0

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation. We started by combining like terms, then isolated the variable, and finally checked our solution to make sure it was correct. By following these steps, we were able to solve for $a$ in the equation $6a + 5a = -11$ .

Real-World Applications

Solving linear equations is a crucial skill in many real-world applications, including:

Physics: Linear equations are used to describe the motion of objects under the influence of forces.
Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Linear equations are used to model economic systems and make predictions about future.

Tips and Tricks

Here are some tips and tricks to help you solve linear equations:

Use a systematic approach: When solving a linear equation, use a systematic approach to combine like terms and isolate the variable.
Check your solution: Always check your solution to make sure it is correct.
Use technology: If you are having trouble solving a linear equation, try using technology, such as a calculator or computer software, to help you solve it.

Common Mistakes

Here are some common mistakes to avoid when solving linear equations:

Not combining like terms: Failing to combine like terms can make it difficult to isolate the variable.
Not checking the solution: Failing to check the solution can lead to incorrect answers.
Not using a systematic approach: Failing to use a systematic approach can make it difficult to solve the equation.

Conclusion

Introduction

In our previous article, we discussed the step-by-step process of solving linear equations. However, we know that practice makes perfect, and sometimes, it's helpful to have a Q&A guide to clarify any doubts or questions you may have. In this article, we will provide a Q&A guide to help you better understand how to solve linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (in this case, $a$ ) is 1. For example, $2x + 3 = 5$ is a linear equation.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients (the numbers in front of the variable) of the terms that have the same variable. For example, in the equation $2x + 3x = 5$ , you can combine the like terms by adding the coefficients: $2x + 3x = 5x$ .

Q: How do I isolate the variable?

A: To isolate the variable, you need to get the variable by itself on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value. For example, in the equation $2x + 3 = 5$ , you can isolate the variable by subtracting 3 from both sides: $2x = 5 - 3$ , which simplifies to $2x = 2$ .

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. For example, $2x + 3 = 5$ is a linear equation, while $x^2 + 2x + 1 = 0$ is a quadratic equation.

Q: How do I check my solution?

A: To check your solution, you need to plug the value of the variable back into the original equation and check if it is true. For example, if you solve the equation $2x + 3 = 5$ and get $x = 1$ , you can plug $x = 1$ back into the original equation to check if it is true: $2(1) + 3 = 5$ , which simplifies to $5 = 5$ , so the solution is correct.

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

A: Some common mistakes to avoid when solving linear equations include:

Not combining like terms
Not checking the solution
Not using a systematic approach
Not using technology to help solve the equation

Q: How can I use technology to help solve linear equations?

A: There are many tools and software available that can help you solve linear equations, including:

Calculators
Computer software, such as Mathematica or Maple
Online equation solvers
Mobile apps, such as Photomath or Mathway

Q: What are some real-world of linear equations?

A: Linear equations have many real-world applications, including:

Physics: Linear equations are used to describe the motion of objects under the influence of forces.
Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Linear equations are used to model economic systems and make predictions about future.

Conclusion

Solving linear equations is a crucial skill in mathematics, and it has many real-world applications. By following the step-by-step process outlined in this article, you can solve linear equations with confidence. Remember to combine like terms, isolate the variable, and check your solution to make sure it is correct. With practice and patience, you will become proficient in solving linear equations and be able to apply this skill to a wide range of problems.