Justifying Solution Steps: 3(x-5) + 7x = 65 - Find The Table!

Let's dive into solving equations and, more importantly, understanding the reasoning behind each step. In mathematics, it's not enough to just arrive at the answer; you need to be able to justify how you got there. This is where properties of equality and other mathematical principles come into play. We'll break down the equation 3(x-5) + 7x = 65, step-by-step, and identify the correct justifications.

Understanding the Equation: 3(x-5) + 7x = 65

Before we jump into the solution, let's get familiar with the equation itself. We have a linear equation in one variable, 'x'. Our goal is to isolate 'x' on one side of the equation to find its value. To do this, we'll use a series of algebraic manipulations, each justified by a specific mathematical property. The equation presented is 3(x-5) + 7x = 65, and our task is to identify the table that accurately justifies each step in solving for 'x'. This means we need to not only solve the equation but also understand the why behind each transformation. We'll be looking for the correct application of properties like the distributive property, combining like terms, and properties of equality (addition, subtraction, multiplication, division). So, let's put on our thinking caps and get ready to dissect this equation!

Keywords: linear equation, properties of equality, distributive property, combining like terms, solving for x, justifying steps

Step-by-Step Solution and Justifications

Let's walk through the solution process, paying close attention to the justification for each step:

1. Original Equation: 3(x - 5) + 7x = 65
2. Step 1: Apply the Distributive Property: This is where we multiply the 3 by both terms inside the parentheses: 3 * x and 3 * -5. This gives us 3x - 15 + 7x = 65. The justification here is the Distributive Property. This property states that a(b + c) = ab + ac.
3. Step 2: Combine Like Terms: We have two terms with 'x' in them: 3x and 7x. We can combine these to get 10x. The equation now looks like this: 10x - 15 = 65. The justification for this step is Combining Like Terms. Like terms are terms that have the same variable raised to the same power, and we can add or subtract their coefficients.
4. Step 3: Isolate the Variable Term: To get the '10x' term by itself, we need to get rid of the '-15'. We do this by adding 15 to both sides of the equation. This gives us 10x - 15 + 15 = 65 + 15, which simplifies to 10x = 80. The justification here is the Addition Property of Equality. This property states that you can add the same value to both sides of an equation without changing the equality.
5. Step 4: Solve for x: Now we have 10x = 80. To isolate 'x', we need to divide both sides of the equation by 10. This gives us 10x / 10 = 80 / 10, which simplifies to x = 8. The justification for this step is the Division Property of Equality. This property states that you can divide both sides of an equation by the same non-zero value without changing the equality.

Keywords: Distributive Property, Combining Like Terms, Addition Property of Equality, Division Property of Equality, isolating the variable, algebraic manipulations

Identifying the Correct Table

Now, let's consider the table presented in the original problem. We need to compare the steps in our solution with the justifications provided in the table to determine if it's accurate. The crucial point is to match each solution step with the correct mathematical principle that justifies it. Any mismatch indicates that the table is incorrect.

Based on our step-by-step breakdown, here's what the correct table should look like:

So, to answer the question