Justification For Step 1 In Solving -22-x=5+6x+9 Combining Like Terms

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The initial equation presented is:

$ -22 - x = 5 + 6x + 9 $

Step 1 of the solution process simplifies the equation to:

$ -22 - x = 14 + 6x $

To understand the justification for this step, we need to carefully examine the transformation that occurred. We'll dissect the equation and pinpoint the mathematical principle that allows us to move from the initial form to the simplified form. This involves exploring the fundamental properties of equality and the rules of algebraic manipulation. Let's embark on this journey of mathematical discovery!

Identifying the Core Principle: Combining Like Terms

The key to understanding Step 1 lies in the concept of combining like terms. In algebraic expressions, like terms are those that have the same variable raised to the same power. Constants (numbers without variables) are also considered like terms. Combining like terms involves adding or subtracting the coefficients (the numerical part of the term) of these terms while keeping the variable part the same. This process simplifies the expression without changing its value.

Looking at the right side of the original equation, $ 5 + 6x + 9 $, we can identify two like terms: the constants 5 and 9. These terms do not have any variables associated with them, making them constants and thus like terms. To combine them, we simply add their values:

$ 5 + 9 = 14 $

This explains how the right side of the equation transforms from $ 5 + 6x + 9 $ to $ 14 + 6x $. The term $ 6x $ remains unchanged because it does not have any other like terms to combine with on the right side of the equation.

The left side of the equation, $ -22 - x $, does not have any like terms to combine at this stage. The term $ -22 $ is a constant, and the term $ -x $ has a variable. Since they are not like terms, they cannot be combined.

Therefore, the justification for Step 1 is the application of the principle of combining like terms. This fundamental algebraic technique allows us to simplify expressions and make equations easier to solve.

Why Other Options Don't Fit

To further solidify our understanding, let's examine why the other options provided are not the correct justification for Step 1:

  • A. The subtraction property of equality: This property states that if you subtract the same value from both sides of an equation, the equation remains balanced. While this property is crucial in solving equations, it's not the operation performed in Step 1. We are not subtracting anything from both sides; we are simplifying one side by combining like terms.
  • B. The multiplication property of equality: This property states that if you multiply both sides of an equation by the same non-zero value, the equation remains balanced. Similar to the subtraction property, this is a valid algebraic manipulation, but it's not what's happening in Step 1. There is no multiplication being performed on both sides of the equation.
  • D. The addition property of equality: This property states that if you add the same value to both sides of an equation, the equation remains balanced. While addition is involved in combining the like terms 5 and 9, the addition property of equality refers to adding the same value to both sides of the equation, which is not what occurred in Step 1.

Therefore, the only option that accurately describes the justification for Step 1 is C. combining like terms.

The Significance of Combining Like Terms

Combining like terms is not just a superficial simplification; it's a crucial step in solving algebraic equations. Here's why:

  • Simplification: By combining like terms, we reduce the complexity of the equation, making it easier to visualize and manipulate. A simpler equation is less prone to errors in subsequent steps.
  • Isolation of the variable: Combining like terms often brings us closer to isolating the variable we are trying to solve for. By simplifying the equation, we can more easily identify the operations that need to be reversed to get the variable alone on one side.
  • Efficiency: Combining like terms streamlines the solution process. It reduces the number of steps required to solve the equation, saving time and effort.
  • Clarity: A simplified equation is easier to understand and interpret. It allows us to see the relationships between the variables and constants more clearly.

In the given equation, combining the constants 5 and 9 on the right side made the equation more manageable. It sets the stage for the next steps in solving for x, which might involve using the addition or subtraction property of equality to isolate the variable terms.

A Broader Perspective: Algebraic Simplification

Combining like terms is a fundamental aspect of algebraic simplification. Simplification, in general, is the process of rewriting an expression or equation in a simpler form without changing its value. This can involve various techniques, including:

  • Combining like terms: As we've discussed, this involves adding or subtracting the coefficients of terms with the same variable and exponent.
  • Distributive property: This property allows us to multiply a factor across terms inside parentheses. For example, $ a(b + c) = ab + ac $.
  • Factoring: This involves expressing an expression as a product of its factors. For example, $ x^2 + 2x + 1 $ can be factored as $ (x + 1)(x + 1) $.
  • Order of operations (PEMDAS/BODMAS): Following the correct order of operations (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) ensures that expressions are evaluated consistently.

Mastering these simplification techniques is essential for success in algebra and higher-level mathematics. They allow us to manipulate expressions and equations effectively, solve problems, and gain deeper insights into mathematical relationships.

Back to the Example: The Next Steps

Now that we've justified Step 1, let's briefly consider what the next steps in solving the equation might be. After simplifying the equation to:

$ -22 - x = 14 + 6x $

our goal is to isolate the variable x. This typically involves:

  1. Moving all terms with x to one side of the equation: We could add x to both sides to eliminate the x term on the left side, resulting in:

    $ -22 = 14 + 7x $

  2. Moving all constant terms to the other side of the equation: We could subtract 14 from both sides to isolate the term with x:

    $ -36 = 7x $

  3. Isolating x by dividing both sides by its coefficient: Finally, we would divide both sides by 7 to solve for x:

    $ x = -36/7 $

These subsequent steps involve applying the addition and subtraction properties of equality, which, as we discussed earlier, are different from the justification for Step 1, which was combining like terms.

Conclusion: A Foundation for Algebraic Success

In conclusion, the justification for Step 1 in the solution process of the equation $ -22 - x = 5 + 6x + 9 $ is C. combining like terms. This fundamental algebraic technique allows us to simplify expressions by adding or subtracting the coefficients of like terms. Understanding this principle, along with other simplification techniques and properties of equality, is crucial for success in algebra and beyond. It's a cornerstone of mathematical problem-solving and a key to unlocking more complex mathematical concepts.

By carefully analyzing each step in the solution process and understanding the underlying principles, we can develop a deeper appreciation for the elegance and power of mathematics. Combining like terms may seem like a simple operation, but it's a vital tool in our mathematical toolkit, enabling us to transform complex expressions into manageable forms and ultimately solve for unknown variables.

  • Combining like terms
  • Algebraic simplification
  • Properties of equality
  • Solving equations
  • Mathematical principles
  • Step-by-step solution
  • Justification for steps
  • Algebra
  • Mathematics
  • Equation simplification