Question

khadija is blocking off several rooms in a hotel for guests coming to her wedding. the hotel can reserve small rooms that can hold 3 people, and large rooms that can hold 4 people. khadija reserved 2 more small rooms than large rooms, which altogether can accommodate 48 guests. graphically solve a system of equations in order to determine the number of small rooms reserved, x, and the number of large rooms reserved, y. click twice to plot each line. click a line to delete it.

Explanation:

Step1: Define variables & equations

Let $x$ = small rooms, $y$ = large rooms.

1. Room count relationship: $x = y + 2$
2. Guest capacity relationship: $3x + 4y = 48$

Step2: Rewrite for graphing

Rewrite equations in slope-intercept form ($y=mx+b$):

1. $x = y + 2 \implies y = x - 2$
2. $3x + 4y = 48 \implies 4y = -3x + 48 \implies y = -\frac{3}{4}x + 12$

Step3: Find intersection (solution)

Substitute $y = x - 2$ into $3x + 4y = 48$:
$3x + 4(x - 2) = 48$
$3x + 4x - 8 = 48$
$7x = 56 \implies x = 8$
Then $y = 8 - 2 = 6$

To graph:

• For $y = x - 2$: Plot points $(2,0)$ and $(8,6)$
• For $y = -\frac{3}{4}x + 12$: Plot points $(16,0)$ and $(8,6)$

The lines intersect at $(8,6)$.

Answer:

Number of small rooms $x = 8$, number of large rooms $y = 6$
(Graph: Two lines intersecting at the coordinate (8, 6))