Question

5 write an equation to model the statement the product of -8 and (y + 3) is 32. how can you use the equation to reason about the value of y?
6 jiro buys 2 rocks that each cost the same amount, r, and a magnifying glass that costs $5. the total cost is $9. model this situation with an equation and a hanger diagram.

Explanation:

Question 5

Step1: Translate the statement into an equation

The product of \(-8\) and \((y + 3)\) is \(32\), so the equation is \(-8(y + 3)=32\).

Step2: Solve for \(y\)

First, divide both sides of the equation by \(-8\): \(\frac{-8(y + 3)}{-8}=\frac{32}{-8}\), which simplifies to \(y + 3=-4\).
Then, subtract \(3\) from both sides: \(y + 3-3=-4-3\), so \(y=-7\).

Step1: Model the situation with an equation

Jiro buys \(2\) rocks at cost \(r\) each and a magnifying glass for \(\$5\), with total cost \(\$9\). So the equation is \(2r + 5 = 9\).

Step2: Model with a hanger diagram (description)

• On one side of the hanger, we have two equal - length segments representing the cost of each rock (\(r\)) and a segment representing the cost of the magnifying glass (\(5\)).
• On the other side of the hanger, we have a segment representing the total cost (\(9\)).

To solve the equation:
Subtract \(5\) from both sides: \(2r+5 - 5=9 - 5\), so \(2r = 4\).
Then divide both sides by \(2\): \(\frac{2r}{2}=\frac{4}{2}\), so \(r = 2\).

Answer:

The equation is \(-8(y + 3)=32\), and the value of \(y\) is \(-7\).

Question 6