Solve The Equation:${ \begin{aligned} -1 & = \frac{5 + X}{6} \ -91 \end{aligned} }$

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific linear equation, step by step, and provide a clear understanding of the process involved.

The Equation

The given equation is:

$$\begin{aligned} -1 & = \frac{5 + x}{6} \\ -91 \end{aligned}$$

This equation represents a linear relationship between the variables x and the constant term. Our goal is to isolate the variable x and find its value.

Step 1: Multiply Both Sides by the Denominator

To eliminate the fraction, we need to multiply both sides of the equation by the denominator, which is 6.

$$\begin{aligned} -1 \times 6 & = \frac{5 + x}{6} \times 6 \\ -6 & = 5 + x \end{aligned}$$

Step 2: Subtract 5 from Both Sides

Next, we need to isolate the variable x by subtracting 5 from both sides of the equation.

$$\begin{aligned} -6 - 5 & = 5 + x - 5 \\ -11 & = x \end{aligned}$$

Step 3: Check the Solution

To verify our solution, we can plug the value of x back into the original equation and check if it holds true.

$$\begin{aligned} \frac{5 + (-11)}{6} & = -1 \\ \frac{-6}{6} & = -1 \\ -1 & = -1 \end{aligned}$$

As expected, the equation holds true, and our solution is correct.

Conclusion

Solving linear equations requires a step-by-step approach, and it's essential to follow the correct order of operations. By multiplying both sides by the denominator, subtracting the constant term, and checking the solution, we can confidently solve linear equations and find the value of the variable.

Common Mistakes to Avoid

When solving linear equations, it's easy to make mistakes. Here are some common pitfalls to avoid:

• Not multiplying both sides by the denominator: Failing to eliminate the fraction can lead to incorrect solutions.
• Not subtracting the constant term: Neglecting to isolate the variable x can result in incorrect answers.
• Not checking the solution: Failing to verify the solution can lead to incorrect conclusions.

Real-World Applications

Linear equations have numerous real-world applications, including:

• Physics and Engineering: Linear equations are used to model physical systems, such as motion and energy.
• Economics: Linear equations are used to model economic systems, such as supply and demand.
• Computer Science: Linear equations are used in algorithms and data structures, such as linear programming.

Tips and Tricks

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

• Use the distributive property: When multiplying both sides by the denominator, use the distributive property to simplify the equation.
• Use inverse operations: When isolating the variable x, use inverse operations, such as addition and subtraction, to simplify the equation.
• Check your work: Always verify your solution by plugging the value of x back into the original equation.

Practice Problems

Here are some practice problems to help you reinforce your understanding of solving linear equations:

• Problem 1: Solve the equation $\begin{aligned} 2x + 3 & = 7 \end{aligned}$
• Problem 2: Solve the equation $\begin{aligned} x - 2 & = 4 \end{aligned}$
• Problem 3: Solve the equation $\begin{aligned} 3x + 2 & = 11 \end{aligned}$

Conclusion

Introduction

In our previous article, we covered the basics of solving linear equations, including step-by-step solutions and common mistakes to avoid. In this article, we'll take a Q&A approach to provide additional insights and clarify any doubts you may have.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.

Q: How do I know if an equation is linear?

A: To determine if an equation is linear, look for the following characteristics:

• The highest power of the variable(s) is 1.
• The equation can be written in the form ax + b = c.
• The equation does not contain any squared or higher powers of the variable(s).

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(s) is 1, whereas a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example:

• Linear equation: 2x + 3 = 7
• Quadratic equation: x^2 + 4x + 4 = 0

Q: How do I solve a linear equation with fractions?

A: To solve a linear equation with fractions, follow these steps:

1. Multiply both sides of the equation by the denominator to eliminate the fraction.
2. Simplify the equation by combining like terms.
3. Isolate the variable by adding or subtracting the constant term.

Q: What is the order of operations when solving linear equations?

A: When solving linear equations, follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I check my solution to a linear equation?

A: To check your solution to a linear equation, plug the value of the variable back into the original equation and simplify. If the equation holds true, then your 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 multiplying both sides by the denominator to eliminate fractions.
• Not simplifying the equation by combining like terms.
• Not isolating the variable by adding or subtracting the constant term.
• Not checking the solution by plugging the value of the variable back into the original equation.

Q: How do I apply linear equations to real-world problems?

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

• Physics and Engineering: Linear equations are used to model physical systems, such as motion and energy.
• Economics: Linear equations are used to model economic systems, such as supply and demand.
• Computer Science: Linear equations are used in algorithms and data structures, such as linear programming.

Conclusion

Solving linear equations is a fundamental skill that requires practice and patience. By following the correct order of operations, avoiding common mistakes, and using real-world applications, you can become proficient in solving linear equations. Remember to check your work and use inverse operations to simplify the equation. With practice, you'll become a master of solving linear equations!

Practice Problems

Here are some practice problems to help you reinforce your understanding of solving linear equations:

• Problem 1: Solve the equation $\begin{aligned} 2x + 3 & = 7 \end{aligned}$
• Problem 2: Solve the equation $\begin{aligned} x - 2 & = 4 \end{aligned}$
• Problem 3: Solve the equation $\begin{aligned} 3x + 2 & = 11 \end{aligned}$

Additional Resources

For more information on solving linear equations, check out the following resources:

• Khan Academy: Linear Equations
• Mathway: Linear Equations
• Wolfram Alpha: Linear Equations

Conclusion

Solving linear equations is a fundamental skill that requires practice and patience. By following the correct order of operations, avoiding common mistakes, and using real-world applications, you can become proficient in solving linear equations. Remember to check your work and use inverse operations to simplify the equation. With practice, you'll become a master of solving linear equations!