Solving Linear Equations How Many Solutions Does 9(x-4) = 9x - 33 Have?

How many solutions are there for 9(x-4)=9x-33?

In the realm of mathematics, solving equations is a fundamental skill. Equations are mathematical statements that assert the equality of two expressions. Finding the solutions to an equation involves determining the values of the variables that make the equation true. In this article, we will delve into the process of solving a linear equation and explore the different possibilities for the number of solutions.

The equation we will be analyzing is:

9(x - 4) = 9x - 33

This is a linear equation in one variable, x. Linear equations are characterized by the fact that the variable appears only to the first power. To solve this equation, we will employ algebraic techniques to isolate the variable x and determine its value.

Unveiling the Solution Process: Step-by-Step

Let's embark on a step-by-step journey to unravel the solution to this equation. Our primary goal is to manipulate the equation using algebraic operations while maintaining its balance. This means that any operation performed on one side of the equation must also be performed on the other side.

1. The Distributive Property: Expanding the Equation

The first step in solving this equation is to apply the distributive property. This property states that for any numbers a, b, and c:

a(b + c) = ab + ac

In our equation, we have 9(x - 4) on the left side. Applying the distributive property, we get:

9(x - 4) = 9 * x - 9 * 4 = 9x - 36

Substituting this back into the original equation, we have:

9x - 36 = 9x - 33

2. Isolating the Variable: Combining Like Terms

Our next objective is to isolate the variable x. To do this, we need to gather all the terms containing x on one side of the equation and all the constant terms on the other side. In this case, we observe that we have 9x on both sides of the equation. Let's subtract 9x from both sides:

9x - 36 - 9x = 9x - 33 - 9x

Simplifying, we get:

-36 = -33

3. The Contradiction: Unveiling the Truth

Now, we arrive at a crucial point. We have the equation -36 = -33. This statement is clearly a contradiction. It asserts that -36 is equal to -33, which is not true. This contradiction reveals a fundamental aspect of the equation.

Interpreting the Contradiction: No Solutions

The contradiction -36 = -33 signifies that there is no value of x that can make the original equation true. In other words, no matter what number we substitute for x, the left side of the equation will never be equal to the right side. Therefore, the equation has no solution.

Graphical Interpretation: Parallel Lines

To gain a deeper understanding of why this equation has no solution, let's consider a graphical interpretation. We can rewrite the original equation as two separate linear equations:

y = 9(x - 4) = 9x - 36
y = 9x - 33

These equations represent two straight lines. The solutions to the original equation correspond to the points where these two lines intersect. The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. In this case, both lines have a slope of 9, but they have different y-intercepts (-36 and -33). This means that the lines are parallel.

Parallel lines, by definition, never intersect. Since the lines representing our equations are parallel, they will never intersect, and there is no point (x, y) that lies on both lines. This confirms our earlier conclusion that the equation has no solution.

Categorizing Equations: Identities, Contradictions, and Conditional Equations

Equations can be classified into three categories based on the number of solutions they possess:

• Identities: Equations that are true for all values of the variable. For example, the equation x + x = 2x is an identity.
• Contradictions: Equations that are false for all values of the variable. Our equation 9(x - 4) = 9x - 33 is a contradiction.
• Conditional Equations: Equations that are true for some values of the variable but false for others. For example, the equation x + 1 = 5 is a conditional equation, as it is only true when x = 4.

Answering the Question: The Final Verdict

Having thoroughly analyzed the equation 9(x - 4) = 9x - 33, we have arrived at the definitive answer. The equation has no solutions. This is because the equation simplifies to a contradiction, -36 = -33, which is a false statement. The graphical interpretation of the equation as two parallel lines further reinforces this conclusion.

Therefore, the correct answer is A. 0

Expanding the Horizon: Further Exploration

This exploration of linear equations and their solutions is just the beginning. The world of mathematics offers a vast landscape of equations, each with its own unique characteristics and solution methods. As you continue your mathematical journey, you will encounter quadratic equations, polynomial equations, trigonometric equations, and many more.

Strategies for Solving Equations

Here are some general strategies that can be applied to solve a wide range of equations:

• Simplify: Begin by simplifying both sides of the equation as much as possible. This may involve using the distributive property, combining like terms, or performing other algebraic manipulations.
• Isolate the Variable: The ultimate goal is to isolate the variable on one side of the equation. This is achieved by performing inverse operations on both sides of the equation. For example, if a term is added to the variable, subtract that term from both sides. If a term is multiplied by the variable, divide both sides by that term.
• Check Your Solution: After you have found a potential solution, it is crucial to check your answer. Substitute the solution back into the original equation to ensure that it makes the equation true.
• Consider Different Solution Methods: Some equations can be solved using multiple methods. Exploring different approaches can provide valuable insights and enhance your problem-solving skills.

Common Pitfalls to Avoid

When solving equations, it is essential to be mindful of common pitfalls that can lead to errors:

• Dividing by Zero: Division by zero is undefined. Avoid dividing both sides of an equation by an expression that could be zero.
• Incorrectly Applying the Distributive Property: Ensure that you distribute correctly when expanding expressions. Pay attention to signs and multiply each term inside the parentheses by the term outside.
• Forgetting to Check Solutions: Always check your solutions to avoid extraneous solutions, which are values that satisfy the transformed equation but not the original equation.
• Making Arithmetic Errors: Careless arithmetic mistakes can lead to incorrect solutions. Double-check your calculations to ensure accuracy.

By mastering these strategies and avoiding common pitfalls, you will be well-equipped to tackle a wide variety of equations.

Conclusion: The Power of Mathematical Inquiry

In this exploration, we have delved into the process of solving a linear equation and discovered that the equation 9(x - 4) = 9x - 33 has no solutions. This journey highlights the power of mathematical inquiry and the importance of rigorous analysis. By understanding the underlying principles and applying algebraic techniques, we can unravel the mysteries of equations and gain deeper insights into the world of mathematics.

Remember, the pursuit of mathematical knowledge is an ongoing adventure. Embrace the challenges, persevere through the complexities, and celebrate the moments of discovery. The world of mathematics awaits, ready to reveal its hidden treasures to those who seek them.

This detailed exploration should give the readers a comprehensive understanding of how to approach such problems and the logic behind arriving at the solution. Remember, practice is the key to mastering these concepts.