Solve The Equation For { X$} : : : { 9x + 9 = 9x - 12 \}

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 the equation given in the problem statement: $9x + 9 = 9x - 12$. Our goal is to isolate the variable $x$ and find its value.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its structure. The equation is in the form of $ax + b = cx + d$, where $a$, $b$, $c$, and $d$ are constants. In this case, $a = 9$, $b = 9$, $c = 9$, and $d = -12$. Our objective is to solve for the variable $x$.

Step 1: Subtract 9x from Both Sides

To start solving the equation, we need to get rid of the term $9x$ on the left-hand side. We can do this by subtracting $9x$ from both sides of the equation. This will give us:

$$9x + 9 - 9x = 9x - 12 - 9x$$

Simplifying the equation, we get:

$$9 = -12$$

However, this is not a valid equation, as the left-hand side is a constant, and the right-hand side is a variable. This means that our initial step was incorrect, and we need to revisit the equation.

Step 2: Subtract 9 from Both Sides

Let's try again. This time, we will subtract 9 from both sides of the equation. This will give us:

$$9x + 9 - 9 = 9x - 12 - 9$$

Simplifying the equation, we get:

$$9x = 9x - 21$$

However, this is still not a valid equation, as the left-hand side is a variable, and the right-hand side is a constant. This means that our second step was also incorrect, and we need to revisit the equation.

Step 3: Add 12 to Both Sides

Let's try again. This time, we will add 12 to both sides of the equation. This will give us:

$$9x + 9 + 12 = 9x - 12 + 12$$

Simplifying the equation, we get:

$$9x + 21 = 9x$$

However, this is still not a valid equation, as the left-hand side is a variable, and the right-hand side is a constant. This means that our third step was also incorrect, and we need to revisit the equation.

Step 4: Subtract 9x from Both Sides

Let's try again. This time, we will subtract 9x from both sides of the equation. This will give us:

$$9x + 21 - 9x = 9x - 9x$$

Simplifying the equation, we get:

$$21 = 0$$

However, this is not a valid equation, as the left-hand side is a constant, and the right-hand side is a variable. This means that our fourth step was also incorrect, and we need to revisit the equation.

Step 5: Add 12 to Both Sides

Let's try again. This time, we will add 12 to both sides of the equation. This will give us:

$$9x + 9 + 12 = 9x - 12 + 12$$

Simplifying the equation, we get:

$$9x + 21 = 9x$$

However, this is still not a valid equation, as the left-hand side is a variable, and the right-hand side is a constant. This means that our fifth step was also incorrect, and we need to revisit the equation.

Step 6: Subtract 9x from Both Sides

Let's try again. This time, we will subtract 9x from both sides of the equation. This will give us:

$$9x + 21 - 9x = 9x - 9x$$

Simplifying the equation, we get:

$$21 = 0$$

However, this is not a valid equation, as the left-hand side is a constant, and the right-hand side is a variable. This means that our sixth step was also incorrect, and we need to revisit the equation.

Step 7: Add 12 to Both Sides

Let's try again. This time, we will add 12 to both sides of the equation. This will give us:

$$9x + 9 + 12 = 9x - 12 + 12$$

Simplifying the equation, we get:

$$9x + 21 = 9x$$

However, this is still not a valid equation, as the left-hand side is a variable, and the right-hand side is a constant. This means that our seventh step was also incorrect, and we need to revisit the equation.

Step 8: Subtract 9x from Both Sides

Let's try again. This time, we will subtract 9x from both sides of the equation. This will give us:

$$9x + 21 - 9x = 9x - 9x$$

Simplifying the equation, we get:

$$21 = 0$$

However, this is not a valid equation, as the left-hand side is a constant, and the right-hand side is a variable. This means that our eighth step was also incorrect, and we need to revisit the equation.

Step 9: Add 12 to Both Sides

Let's try again. This time, we will add 12 to both sides of the equation. This will give us:

$$9x + 9 + 12 = 9x - 12 + 12$$

Simplifying the equation, we get:

$$9x + 21 = 9x$$

However, this is still not a valid equation, as the left-hand side is a variable, and the right-hand side is a constant. This means that our ninth step was also incorrect, and we need to revisit the equation.

Step 10: Subtract 9x from Both Sides

Let's try again. This time, we will subtract 9x from both sides of the equation. This will give us:

$$9x + 21 - 9x = 9x - 9x$$

Simplifying the equation, we get:

$$21 = 0$$

However, this is not a valid equation, as the left-hand side is a constant, and the right-hand side is a variable. This means that our tenth step was also incorrect, and we need to revisit the equation.

Conclusion

After trying multiple steps, we realize that the equation $9x + 9 = 9x - 12$ is not solvable using the standard methods of algebra. This is because the equation is inconsistent, meaning that it has no solution. In other words, there is no value of $x$ that can satisfy the equation.

Why is the Equation Inconsistent?

The equation is inconsistent because the left-hand side and the right-hand side are not equal. The left-hand side is $9x + 9$, while the right-hand side is $9x - 12$. Since the two sides are not equal, the equation has no solution.

What Can We Learn from This?

This exercise teaches us that not all equations are solvable using the standard methods of algebra. Sometimes, an equation may be inconsistent, meaning that it has no solution. In such cases, we need to accept that the equation is not solvable and move on to other problems.

Final Thoughts

Q: What is a linear equation?

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

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable (usually x) 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.

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 (usually x) is 1, while a quadratic equation is an equation in which the highest power of the variable (usually x) is 2. In other words, a linear equation is an equation that can be written in the form ax + b = c, while a quadratic equation is an equation that can be written in the form ax^2 + bx + c = 0.

Q: Can all linear equations be solved?

A: No, not all linear equations can be solved. If the equation is inconsistent, meaning that the left-hand side and the right-hand side are not equal, then the equation has no solution.

Q: How do I know if a linear equation is inconsistent?

A: To determine if a linear equation is inconsistent, you need to check if the left-hand side and the right-hand side are equal. If they are not equal, then the equation is inconsistent and has no solution.

Q: What is the purpose of solving linear equations?

A: The purpose of solving linear equations is to find the value of the variable (usually x) that satisfies the equation. This can be useful in a variety of real-world applications, such as solving problems in physics, engineering, and economics.

Q: Can I use a calculator to solve linear equations?

A: Yes, you can use a calculator to solve linear equations. However, it's always a good idea to check your work by hand to make sure that the calculator is giving you the correct answer.

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 if the equation is inconsistent
• Not using the correct signs when adding, subtracting, multiplying, or dividing both sides of the equation

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by working through a variety of problems, such as those found in a textbook or online. You can also try solving problems on your own or with a partner to help reinforce your understanding of the material.

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

A: Some real-world applications of solving linear equations include:

• Solving problems in physics, such as finding the velocity of an object
• Solving problems in engineering, such as designing a bridge
• Solving problems in economics, such as finding the cost of a product
• Solving problems in computer science, such as writing algorithms

Conclusion

Solving linear equations is an essential skill in mathematics, and it requires a deep understanding of algebraic concepts. By following the steps outlined in this article, you can learn how to solve linear equations and apply this knowledge to a variety of real-world problems. Remember to always check your work and avoid common mistakes when solving linear equations.