Question

1. a hotel charges a fee for renting its banquet hall and an additional price per guest. the expression 18x + 500 represents the total price, in dollars, that the hotel charges. interpret what each part of the expression represents.

x:
18x:
18:
500:
objective: understand the standard form of a linear equation and use intercepts to graph a line.

1. the glee club has $90 to spend on pens and pencils. each pen costs $0.75, and each pencil costs $0.15. let x represent the number of pens, and let y represent the number of pencils.

write an equation describing the number of pens and pencils the glee club can buy.
what is the greatest number of each type of writing tool that the club can buy?
pencils:
pens:

1. deondra has $30 to spend on a mixture of green and red grapes.

a) write an equation to model the amounts of green, x, and red, y, grapes she can buy for $30.
b) find the x and y-intercepts to determine the possible combinations of green and red grapes she can buy. provide a combination.
c) graph the intercepts and determine the possible number of combinations of red and green graphs deondra can buy.

1. rewrite the equation into slope-intercept form.

2x + 3y = 4

1. rewrite the equation into slope-intercept form.

3x - 2y = -5

Explanation:

Step1: Interpret variable x

$x$ = number of guests at the banquet

Step2: Interpret term 18x

$18x$ = total cost for all guests

Step3: Interpret coefficient 18

$18$ = cost per guest (in dollars)

Step4: Interpret constant 500

$500$ = fixed banquet hall rental fee (in dollars)

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Step1: Write total cost equation

$0.75x + 0.15y = 90$

Step2: Maximize pencils (set x=0)

$0.15y = 90 \implies y = \frac{90}{0.15} = 600$

Step3: Maximize pens (set y=0)

$0.75x = 90 \implies x = \frac{90}{0.75} = 120$

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Step1: Write grape cost equation

$3x + 2y = 30$

Step2: Find x-intercept (y=0)

$3x = 30 \implies x = 10$ (10 lbs green grapes, 0 red)

Step3: Find y-intercept (x=0)

$2y = 30 \implies y = 15$ (0 lbs green grapes, 15 red)

Step4: Example combination

Let $x=4$: $3(4)+2y=30 \implies 12+2y=30 \implies y=9$ (4 lbs green, 9 lbs red)

Step5: Graph intercepts

Plot points (10, 0) and (0, 15), draw a line between them. All non-negative integer pairs $(x,y)$ on/under the line are valid combinations.

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Step1: Isolate y for 2x+3y=4

$3y = -2x + 4 \implies y = -\frac{2}{3}x + \frac{4}{3}$

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Step1: Isolate y for 3x-2y=-5

$-2y = -3x -5 \implies y = \frac{3}{2}x + \frac{5}{2}$

Answer:

Problem 11

• $x$: Number of banquet guests
• $18x$: Total cost for all guests
• $18$: Cost per guest (in dollars)
• $500$: Fixed hall rental fee (in dollars)

Problem 12

• Equation: $0.75x + 0.15y = 90$
• Pencils: 600
• Pens: 120

Problem 13

a) $3x + 2y = 30$
b) x-intercept: (10, 0) (10 lbs green, 0 red); y-intercept: (0, 15) (0 lbs green, 15 red); Example combination: 4 lbs green grapes and 9 lbs red grapes
c) Plot (10, 0) and (0, 15), draw a line connecting them. All non-negative integer pairs $(x,y)$ that satisfy $3x+2y\leq30$ are valid combinations.

Problem 14

$y = -\frac{2}{3}x + \frac{4}{3}$

Problem 15

$y = \frac{3}{2}x + \frac{5}{2}$