Question

1. which uses the gcf to generate an expression equivalent to \\(\frac{10}{7}x - \frac{5}{7}\\)?

a \\(\frac{5}{7}(2x - 1)\\)
b \\(\frac{5}{7}(2x + 1)\\)
c \\(\frac{1}{7}(10x - 1)\\)
d \\(\frac{1}{7}(10x + 1)\\)

1. megan’s room was remodeled. the new area of the room is 175% of the previous area. only the length of the room changed.

image of a square with side 4 m and an extension of length x
by how many square meters has the area of megan’s room increased? explain.
the area of megan’s room increased by 4, 12, 16, 28 m².
the original area was 4, 12, 16, 28 m²,
75% of 4, 12, 16, 28 is 4, 12, 16, 28.

Explanation:

Question 10 (Subfield: Algebra)

Step1: Find GCF of coefficients

The terms are $\frac{10}{7}x$ and $-\frac{5}{7}$. The GCF of 10 and 5 (numerators) is 5, and the denominator is 7. So GCF is $\frac{5}{7}$.

Step2: Factor out GCF

Factor $\frac{5}{7}$ from $\frac{10}{7}x - \frac{5}{7}$: $\frac{5}{7}(2x - 1)$ (since $\frac{5}{7} \times 2x = \frac{10}{7}x$ and $\frac{5}{7} \times (-1) = -\frac{5}{7}$).

Step1: Find original area

Original room is a square with side 4 m, so area $A_{original} = 4 \times 4 = 16$ $m^2$.

Step2: Find new area

New area is 175% of original: $1.75 \times 16 = 28$ $m^2$.

Step3: Find increase in area

Increase = New area - Original area = $28 - 16 = 12$ $m^2$. Also, 75% of original area: $0.75 \times 16 = 12$ $m^2$ (matches the increase).

Answer:

A. $\frac{5}{7}(2x - 1)$

Question 11 (Subfield: Algebra/Geometry)