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To solve the equation: 24(34+x)=1,200 , give the reasons for each of the given steps?

What was done to the previous step to get to the current equation?

Step 1: 816+24x=1,200

Step 2: 816−816+24x=1,200 − 816

Step 3: 24x=384

Step 4: x=16

3 answers

To analyze the steps taken to solve the equation \( 24(34 + x) = 1,200 \) and provide reasons for each step, we can break them down as follows:

Step 1: 816 + 24x = 1,200
Reason: This step involves distributing \( 24 \) across the expression \( (34 + x) \). Specifically, \( 24 \times 34 \) equals \( 816 \) and \( 24 \times x \) equals \( 24x \). Therefore, after distribution, we rewrite the equation as \( 816 + 24x = 1,200 \).

Step 2: 816 - 816 + 24x = 1,200 - 816
Reason: This step is applying the property of equality, which states that if you subtract the same number from both sides of an equation, the two sides remain equal. Here, \( 816 \) is subtracted from both sides to isolate the term containing \( x \). The left side simplifies to \( 24x \), while the right side simplifies to \( 384 \).

Step 3: 24x = 384
Reason: This step simplifies the equation from Step 2. After performing the subtractions, we state the resulting equation, which is now \( 24x = 384 \).

Step 4: x = 16
Reason: Here, we divide both sides of the equation by \( 24 \) to isolate \( x \). Dividing \( 384 \) by \( 24 \) yields \( 16 \), resulting in the solution \( x = 16 \).

In summary:

  1. Distribution (Step 1): The multiplication of \( 24 \) across \( (34 + x) \).
  2. Subtraction (Step 2): Equal subtraction of \( 816 \) from both sides to isolate \( x \).
  3. Simplification (Step 3): Resulting equation from the previous step.
  4. Division (Step 4): Isolating \( x \) by dividing both sides by \( 24 \).
:: Addition as the inverse operation to isolate the variable

:: Multiplication as the inverse operation to isolate the variable

: Distributive Property

:: Subtraction as the inverse operation to isolate the variable

:: Combine Like Terms

#: Division as the inverse operation to isolate the variable

It appears you're looking for the correct matches of reasoning for each step based on the operations involved in solving the equation. Here's how you can align the operations to each step:

Review of Steps with Reasoning

  1. Step 1: \( 816 + 24x = 1,200 \)
    Reason: Distributive Property

    • This step uses the Distributive Property to expand \( 24(34 + x) \) into \( 24 \cdot 34 + 24 \cdot x = 816 + 24x \).

  2. Step 2: \( 816 - 816 + 24x = 1,200 - 816 \)
    Reason: Subtraction as the inverse operation to isolate the variable

    • Subtracting \( 816 \) from both sides to isolate the \( 24x \) term.

  3. Step 3: \( 24x = 384 \)
    Reason: Combine Like Terms

    • This step results from simplifying the equation after applying the subtraction from the previous step, combining constants on the right side.

  4. Step 4: \( x = 16 \)
    Reason: Division as the inverse operation to isolate the variable

    • Dividing both sides by \( 24 \) to solve for \( x \).

Summary of Corresponding Reasons

  • Step 1: Distributive Property
  • Step 2: Subtraction as the inverse operation to isolate the variable
  • Step 3: Combine Like Terms
  • Step 4: Division as the inverse operation to isolate the variable

Non-corresponding Statements

  • Addition as the inverse operation and Multiplication as the inverse operation are not applicable to the steps provided, as they don't appear in the progression of steps to isolate \( x \).

Feel free to ask if you need further clarification or additional explanations!