Question

1. lin and noah are solving the equation 7(x + 2) = 91.

a. show what lin’s and noah’s full solution methods might look like. lin starts by using the distributive property. noah starts by dividing each side by 7.
b. what is the same and what is different about their methods?

Explanation:

Part a: Lin's Solution (Using Distributive Property)

Step 1: Apply Distributive Property

The equation is \( 7(x + 2) = 91 \). Using the distributive property \( a(b + c)=ab + ac \), we get:
\( 7x + 7\times2 = 91 \)
Simplify \( 7\times2 \):
\( 7x + 14 = 91 \)

Step 2: Subtract 14 from Both Sides

Subtract 14 from each side to isolate the term with \( x \):
\( 7x + 14 - 14 = 91 - 14 \)
Simplify:
\( 7x = 77 \)

Step 3: Divide by 7

Divide both sides by 7 to solve for \( x \):
\( \frac{7x}{7} = \frac{77}{7} \)
Simplify:
\( x = 11 \)

Part a: Noah's Solution (Dividing by 7 First)

Step 1: Divide Both Sides by 7

The equation is \( 7(x + 2) = 91 \). Divide both sides by 7:
\( \frac{7(x + 2)}{7} = \frac{91}{7} \)
Simplify:
\( x + 2 = 13 \)

Step 2: Subtract 2 from Both Sides

Subtract 2 from each side to solve for \( x \):
\( x + 2 - 2 = 13 - 2 \)
Simplify:
\( x = 11 \)

Part b: Comparing Methods

• Similarities: Both use algebraic operations (subtraction, division) to isolate \( x \) and find the same solution \( x = 11 \).
• Differences: Lin expands the left - hand side using the distributive property before isolating \( x \), while Noah simplifies the equation by dividing both sides by 7 at the beginning. This shows two valid but distinct approaches to solving the linear equation.

Answer:

\( x = 11 \)