Solve For $x$: $9x + 16 = 6x + 1$A. $x = -4$ B. $x = 3$ C. $x = -5$

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, which is a first-degree equation in one variable. We will use the given equation $9x + 16 = 6x + 1$ as an example to demonstrate the step-by-step process of solving for $x$.

What is a Linear Equation?

A linear equation is an equation in which the highest power of the variable (in this case, $x$) is 1. It can be written in the general form $ax + b = c$, where $a$, $b$, and $c$ are constants. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.

The Given Equation

The given equation is $9x + 16 = 6x + 1$. Our goal is to solve for $x$, which means we need to isolate the variable $x$ on one side of the equation.

Step 1: Subtract $6x$ from Both Sides

To start solving the equation, we need to get all the terms involving $x$ on one side of the equation. We can do this by subtracting $6x$ from both sides of the equation.

$$9x + 16 = 6x + 1$$

Subtracting $6x$ from both sides gives us:

$$3x + 16 = 1$$

Step 2: Subtract 16 from Both Sides

Next, we need to get rid of the constant term on the left-hand side of the equation. We can do this by subtracting 16 from both sides of the equation.

$$3x + 16 = 1$$

Subtracting 16 from both sides gives us:

$$3x = -15$$

Step 3: Divide Both Sides by 3

Finally, we need to isolate the variable $x$ by dividing both sides of the equation by 3.

$$3x = -15$$

Dividing both sides by 3 gives us:

$$x = -5$$

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation. We started with the given equation $9x + 16 = 6x + 1$ and used algebraic manipulation to isolate the variable $x$ on one side of the equation. By following these steps, we were able to solve for $x$ and find the solution $x = -5$.

Why is Solving Linear Equations Important?

Solving linear equations is an essential skill in mathematics, and it has numerous applications in various fields, including science, engineering, economics, and finance. Linear equations are used to model real-world problems, such as population growth, motion, and optimization. By solving linear equations, we can gain insights into the behavior of complex systems and make informed decisions.

Common Mistakes to Avoid

When solving linear equations, it's essential to avoid common mistakes, such as:

• Not following the order of operations (PEMDAS)
• Not isolating the variable on one side of the equation
• Not checking the solution by plugging it back into the original equation

Tips and Tricks

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

• Use algebraic manipulation to simplify the equation
• Use graphing to visualize the solution
• Use substitution to solve for the variable
• Check the solution by plugging it back into the original equation

Conclusion

Introduction

In our previous article, we demonstrated the step-by-step process of solving a linear equation. However, we know that practice makes perfect, and there's no better way to learn than by asking questions and getting answers. In this article, we'll address some common questions and concerns that students often have when it comes to solving linear equations.

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 (in this case, $x$) is 1. On the other hand, a quadratic equation is an equation in which the highest power of the variable is 2. For example, $x^2 + 4x + 4 = 0$ is a quadratic equation, while $2x + 5 = 3$ is a linear equation.

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

A: To determine if an equation is linear or quadratic, look for the highest power of the variable. If the highest power is 1, it's a linear equation. If the highest power is 2, it's a quadratic equation.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. PEMDAS stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next (e.g., $x^2$).
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 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 least common multiple (LCM) of the denominators to eliminate the fractions.
2. Simplify the equation by combining like terms.
3. Solve for the variable using algebraic manipulation.

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is a single equation with one variable. A system of linear equations, on the other hand, is a set of two or more linear equations with the same variables. For example, $2x + 3y = 4$ and $x - 2y = -3$ is a system of linear equations.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, follow these steps:

1. Use the substitution method to solve one equation for one variable.
2. Substitute the expression for the variable into the other equation.
3. Solve for the remaining variable using algebraic manipulation.

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

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

• Not following the order of operations (PEMDAS)
• Not isolating the variable on one side of the equation
• Not checking the solution by plugging it back into the original equation
• Not using algebraic manipulation to simplify the equation

Conclusion

Solving linear equations is a fundamental skill in mathematics, and it's essential to have a solid understanding of the concepts and techniques involved. By asking questions and getting answers, we can clarify any doubts and improve our problem-solving skills. Remember to practice regularly and seek help when needed to become a master of solving linear equations!