Question

5)
$x + 7y = 0$
$2x - 8y = 22$
6)
$4x + 2y = -20$
$-2x - 2y = 10$
7)
$x = -6y - 4$
$-6x - 36y = 24$
8)
$-8x - 10y = 24$
$4x + 5y = 4$

1. mountain view high school is hosting an event for their middle schools, and each school needs to get their students to mvhs. victory middle school will take 1 van and 6 buses for a total of 372 students. lake hazel middle school takes 4 vans and 12 buses for 780 students. write a system of linear equations and solve it to find the number of students in each van and bus.

let $v$ represent the number of students on each van.
let $b$ represent the number of students on each bus.

Explanation:

Problem 5

Step1: Isolate x from first equation

$x = -7y$

Step2: Substitute x into second equation

$2(-7y) - 8y = 22$

Step3: Simplify to solve for y

$-14y -8y = 22 \\
-22y = 22 \\
y = -1$

Step4: Substitute y back to find x

$x = -7(-1) = 7$

Problem 6

Step1: Add the two equations

$(4x + 2y) + (-2x -2y) = -20 + 10$

Step2: Simplify to solve for x

$2x = -10 \\
x = -5$

Step3: Substitute x into second equation

$-2(-5) -2y = 10 \\
10 -2y = 10$

Step4: Solve for y

$-2y = 0 \\
y = 0$

Problem 7

Step1: Substitute x into second equation

$-6(-6y -4) -36y = 24$

Step2: Simplify the left side

$36y +24 -36y = 24 \\
24 = 24$

Step3: Interpret the result

This is a true statement, so there are infinitely many solutions.

Problem 8

Step1: Multiply second equation by 2

$2(4x +5y) = 2(4) \\
8x +10y = 8$

Step2: Add to first equation

$(-8x -10y) + (8x +10y) = 24 +8$

Step3: Simplify and interpret

$0 = 32$
This is a false statement, so there is no solution.

Problem 9

Step1: Write equations from context

Victory School: $v + 6b = 372$
Lake Hazel School: $4v + 12b = 780$

Step2: Simplify second equation

Divide by 4: $v + 3b = 195$

Step3: Subtract simplified equation from first

$(v +6b) - (v +3b) = 372 -195 \\
3b = 177$

Step4: Solve for b

$b = 59$

Step5: Substitute b to find v

$v + 3(59) = 195 \\
v +177 = 195 \\
v = 18$

Answer:

1. $x=7$, $y=-1$
2. $x=-5$, $y=0$
3. Infinitely many solutions
4. No solution
5. System:

$$\begin{cases} v + 6b = 372 \\ 4v + 12b = 780 \end{cases}$$

; 18 students per van, 59 students per bus