Question

multiple choice
for exercises 1 - 5, choose the correct letter.

1. the drama club charges $5 for each adult and $3 for each student to attend the school play. the club earned $1850 for their most recent performance, and there were 500 people in attendance. which system of equations could be used to determine the number of students who attended the play?

a. $5a + 3s = 500$
$a + s = 1850$
b. $3a + 5s = 1850$
$a + s = 500$
c. $5a + 3s = 1850$
$a + s = 500$
d. $3a + 5s = 500$
$a + s = 1850$

1. the perimeter of a rectangle is 24 in. and its length $l$ is 3 times the width $w$. what is the length and width $(l, w)$ of the rectangle?

f. $(3, 9)$
g. $(14.4, 4.8)$
h. $(12, 4)$
j. $(9, 3)$

1. the $x$-intercept of a line is 1.5. its $y$-intercept is $\frac{9}{7}$. a second line has an $x$-intercept of 4.25 and a $y$-intercept of $\frac{17}{3}$. what is the value of $x$ at the point where the two lines intersect?

a. $-6\frac{3}{5}$
b. 2
c. $9\frac{1}{5}$
d. 3

1. which of the following systems is inconsistent?

f. $4x + 8y = 16$
$2x + 4y = 4$
g. $4x + 8y = 16$
$2x + 8y = 8$
h. $4x + 8y = 16$
$x + 2y = 4$
j. $4x + 8y = 16$
$2x + 2y = 8$

1. what is the solution of the system of equations? $5x + 7y = 3$

$2x = -3y + 1$
a. $(11, 17)$
b. $(2, -1)$
c. $(11.5, 8)$
d. $(-2, 1)$
short response

1. a hotel is offering the two weekend specials described below.

plan 1: 3 nights and 4 meals for $233$
plan 2: 3 nights and 3 meals for $226.50$
for accounting purposes, the hotel will record income for the stay and the meals separately. so the cost per night for each special must be the same, and the cost per meal for each special must be the same.
a. what is a system of equations for the situation?
b. solve the system. what is the cost per night and the cost per meal?

Explanation:

Step1: Analyze Problem 1

Let \(a\) be the number of adults and \(s\) be the number of students. The total number of people is \(a + s=500\). The total money earned is \(5a + 3s = 1850\) (since adults pay \(5\) dollars and students pay \(3\) dollars).

Step2: Analyze Problem 2

The perimeter formula for a rectangle is \(P=2(\ell + w)\). Given \(P = 24\) and \(\ell=3w\), substitute into the formula: \(24=2(3w + w)\), \(24 = 8w\), \(w = 3\), \(\ell=9\). The pair \((\ell,w)=(9,3)\)

Step3: Analyze Problem 3

The equation of a line in intercept form is \(\frac{x}{x - intercept}+\frac{y}{y - intercept}=1\).
First line: \(\frac{x}{1.5}+\frac{y}{\frac{9}{7}} = 1\), which simplifies to \(\frac{2x}{3}+\frac{7y}{9}=1\), \(6x + 7y=9\)
Second line: \(\frac{x}{4.25}+\frac{y}{\frac{17}{3}}=1\), \(\frac{4x}{17}+\frac{3y}{17}=1\), \(4x+3y = 17\)
Solve the system \(

$$\begin{cases}6x + 7y=9\\4x+3y = 17\end{cases}$$

\)
Multiply the first equation by \(2\) and the second by \(3\): \(

$$\begin{cases}12x+14y = 18\\12x+9y=51\end{cases}$$

\)
Subtract: \((12x + 14y)-(12x + 9y)=18 - 51\), \(5y=-33\), \(y=-\frac{33}{5}\)
Substitute \(y =-\frac{33}{5}\) into \(6x+7y = 9\): \(6x+7\times(-\frac{33}{5})=9\), \(6x=\frac{231 + 45}{5}=\frac{276}{5}\), \(x=-6\frac{3}{5}\)

Step4: Analyze Problem 4

For a system to be inconsistent, the lines are parallel (same slope).
For \(Ax+By = C\), slope \(m=-\frac{A}{B}\)
For \(4x + 8y=16\) (slope \(m=-\frac{4}{8}=-\frac{1}{2}\)) and \(2x + 4y = 4\) (slope \(m=-\frac{2}{4}=-\frac{1}{2}\)), but \(4x + 8y=16\) and \(2x + 4y = 4\) (multiply the second by \(2\): \(4x+8y = 8\)) are parallel and non - coincident.

Step5: Analyze Problem 5

From \(2x=-3y + 1\), \(x=\frac{-3y + 1}{2}\)
Substitute into \(5x+7y = 3\): \(5\times\frac{-3y + 1}{2}+7y=3\), \(\frac{-15y+5 + 14y}{2}=3\), \(-y+5 = 6\), \(y=-1\)
Substitute \(y = - 1\) into \(x=\frac{-3y + 1}{2}\), \(x = 2\)

Step6: Analyze Problem 6

Let \(x\) be the cost per night and \(y\) be the cost per meal.
a. System: \(

$$\begin{cases}3x + 4y=233\\3x+3y=226.5\end{cases}$$

\)
b. Subtract the second equation from the first: \((3x + 4y)-(3x + 3y)=233 - 226.5\), \(y = 6.5\)
Substitute \(y = 6.5\) into \(3x+3y=226.5\), \(3x+3\times6.5=226.5\), \(3x=226.5 - 19.5\), \(3x=207\), \(x = 69\)

Answer:

1. C. \(5a + 3s = 1850\), \(a + s = 500\)
2. J. \((9,3)\)
3. A. \(-6\frac{3}{5}\)
4. F. \(4x + 8y=16\), \(2x + 4y = 4\)
5. B. \((2,-1)\)
6. a. \(

$$\begin{cases}3x + 4y=233\\3x+3y=226.5\end{cases}$$

\)
b. Cost per night \(x = 69\) dollars, cost per meal \(y = 6.5\) dollars