Ms. Jones is buying raisins and peanuts. She needs 8 pounds total, and she wants to have 1.5 more pounds of peanuts than raisins. The solution to the equation below shows how she can determine the solution for n, the number of pounds of peanuts she should buy.

Question

What explanations can be made for each of the three steps in the procedure?
Tell the correct explanation next to each step.
Given: n + (n - 1.5) = 8
Step 1: 2n - 1.5 = 8 Rational: BLANK
Step 2: 2n = 9.5 Rational: BLANK
Step 3: n = 4.75 Rational: BLANK

What properties go in the BLANK's?

Answer 1

To explain the steps taken to solve the equation \( n + (n - 1.5) = 8 \), we will fill in the rationale for each step based on algebraic properties.

Given: \( n + (n - 1.5) = 8 \)

Step 1: \( 2n - 1.5 = 8 \)
Rational: Combining like terms
Explanation: In this step, Ms. Jones combined the two terms \( n \) and \( n - 1.5 \) on the left-hand side of the equation. The two like terms \( n \) and \( n \) combine to \( 2n \), and the constant \(-1.5\) remains, giving the equation \( 2n - 1.5 = 8 \).

Step 2: \( 2n = 9.5 \)
Rational: Adding \( 1.5 \) to both sides
Explanation: In this step, Ms. Jones isolated the term \( 2n \) by adding \( 1.5 \) to both sides of the equation. This is done to move the constant term to the right-hand side, yielding \( 2n = 8 + 1.5 \) or \( 2n = 9.5 \).

Step 3: \( n = 4.75 \)
Rational: Dividing both sides by \( 2 \)
Explanation: In this step, Ms. Jones solved for \( n \) by dividing both sides of the equation by \( 2 \). This is an application of the division property of equality, which states that if you divide both sides of an equation by the same non-zero number, the two sides remain equal. Thus \( n = \frac{9.5}{2} = 4.75 \).

Putting it all together:

• Step 1: Rational: Combining like terms
• Step 2: Rational: Adding \( 1.5 \) to both sides
• Step 3: Rational: Dividing both sides by \( 2 \)