Question

for mothers day, bob made a pop-up card for his mom. he started with a large paper rectangle. then, bob cut out a square with an area of 81 square inches from the middle of the large rectangle. the area of the piece bob had left was 129 square inches. let r represent the area of the rectangle that bob started with. which equation models the problem? \\( r - 81 = 129 \\) \\( r + 81 = 129 \\) solve this equation to find the area of the rectangle that bob started with. \\( \square \\) square inches

Explanation:

Part 1: Equation Modeling

Step1: Analyze the relationship

The original area of the rectangle is \( r \). After cutting out a square with area 81, the remaining area is 129. So original area minus cut - out area equals remaining area.

Step2: Form the equation

Using the relationship from Step 1, the equation is \( r - 81=129 \).

Part 2: Solving the Equation

Step1: Isolate \( r \)

To solve \( r - 81 = 129 \), we add 81 to both sides of the equation.
\( r-81 + 81=129 + 81 \)

Step2: Calculate the value of \( r \)

Simplifying the left - hand side (LHS) and right - hand side (RHS) of the equation:
LHS: \( r-81 + 81=r \)
RHS: \( 129+81 = 210 \)

Answer:

(Equation): \( r - 81=129 \)