Solve For $x$.$9(x+1) = 25 + X$A. \$x = 2$[/tex\] B. $x = 3$ C. $x = 4$ D. \$x = 5$[/tex\]

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, namely the equation $9(x+1) = 25 + x$. We will break down the solution process into manageable steps, making it easier for readers to understand and follow along.

Understanding the Equation

Before we dive into the solution, let's take a closer look at the equation $9(x+1) = 25 + x$. This equation is a linear equation in one variable, which means it has only one unknown value, denoted by the variable $x$. The equation is also a simple algebraic expression, consisting of a product of two terms on the left-hand side and a sum of two terms on the right-hand side.

Step 1: Distribute the 9

To solve the equation, we need to start by distributing the 9 to the terms inside the parentheses on the left-hand side. This will give us:

$$9(x+1) = 9x + 9$$

Step 2: Simplify the Right-Hand Side

Next, we can simplify the right-hand side of the equation by combining the two terms:

$$25 + x = x + 25$$

Step 3: Set Up the Equation

Now that we have simplified both sides of the equation, we can set up the equation as follows:

$$9x + 9 = x + 25$$

Step 4: Subtract x from Both Sides

To isolate the variable $x$, we need to subtract $x$ from both sides of the equation. This will give us:

$$8x + 9 = 25$$

Step 5: Subtract 9 from Both Sides

Next, we can subtract 9 from both sides of the equation to get:

$$8x = 16$$

Step 6: Divide Both Sides by 8

Finally, we can divide both sides of the equation by 8 to solve for $x$:

$$x = \frac{16}{8}$$

Simplifying the Fraction

The fraction $\frac{16}{8}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 8. This gives us:

$$x = \frac{2}{1}$$

Conclusion

In conclusion, we have solved the linear equation $9(x+1) = 25 + x$ by following a series of steps. We distributed the 9, simplified the right-hand side, set up the equation, subtracted $x$ from both sides, subtracted 9 from both sides, and finally divided both sides by 8 to solve for $x$. The solution to the equation is $x = 2$.

Answer

The correct answer is:

A. $x = 2$

Why is this the correct answer?

This is the correct answer because we solved the equation by following a series of steps, and the final solution is $x = 2$. This means that when we substitute $x = 2$ into the original equation, the equation will be true.

What if we had chosen a different answer?

If we had chosen a different answer, such as $x = 3$ or $x = 4$, the equation would not be true. This is because the solution to the equation is unique, and there is only one value of $x$ that satisfies the equation.

What are some common mistakes to avoid?

When solving linear equations, there are several common mistakes to avoid. These include:

• Not distributing the 9 to the terms inside the parentheses on the left-hand side
• Not simplifying the right-hand side of the equation
• Not setting up the equation correctly
• Not subtracting $x$ from both sides of the equation
• Not subtracting 9 from both sides of the equation
• Not dividing both sides of the equation by 8

By avoiding these common mistakes, we can ensure that we solve the equation correctly and find the unique solution to the equation.

What are some real-world applications of linear equations?

Linear equations have many real-world applications, including:

• Modeling population growth
• Calculating interest rates
• Determining the cost of goods
• Solving problems in physics and engineering

By understanding how to solve linear equations, we can apply this knowledge to a wide range of real-world problems and make informed decisions.

Conclusion

Introduction

In our previous article, we discussed how to solve linear equations, including the equation $9(x+1) = 25 + x$. We broke down the solution process into manageable steps and provided a clear explanation of each step. In this article, we will answer some common questions that students may have when solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. In other words, it is an equation in which the variable is not raised to a power greater than 1.

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

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

• The equation has only one variable.
• The variable is not raised to a power greater than 1.
• The equation does not contain any fractions or decimals.

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:

• Linear equation: $2x + 3 = 5$
• Quadratic equation: $x^2 + 4x + 4 = 0$

Q: How do I solve a linear equation?

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

1. Distribute any coefficients to the terms inside the parentheses.
2. Simplify the right-hand side of the equation.
3. Set up the equation by combining like terms.
4. Subtract the variable from both sides of the equation.
5. Subtract any constants from both sides of the equation.
6. Divide both sides of the equation by the coefficient of the variable.

Q: What if I have a fraction or decimal in my equation?

A: If you have a fraction or decimal in your equation, you can eliminate it by multiplying both sides of the equation by the denominator or by multiplying both sides of the equation by 10, respectively.

Q: What if I have a negative coefficient in my equation?

A: If you have a negative coefficient in your equation, you can eliminate it by multiplying both sides of the equation by -1.

Q: How do I check my solution?

A: To check your solution, substitute the value of the variable back into the original equation and simplify. If the equation is 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 distributing coefficients to the terms inside the parentheses.
• Not simplifying the right-hand side of the equation.
• Not setting up the equation correctly.
• Not subtracting the variable from both sides of the equation.
• Not subtracting any constants from both sides of the equation.
• Not dividing both sides of the equation by the coefficient of the variable.

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

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

• Modeling population growth
• Calculating interest rates
• Determining the cost of goods
• Solving problems in physics and engineering

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following a series of steps and avoiding common mistakes, we can solve equations like $9(x+1) = 25 + x$ and find the unique solution to the equation. We hope that this Q&A guide has been helpful in answering some of the most common questions that students may have when solving linear equations.