Question

section a: linear systems
solve the following systems of equations using the method of your choice. name your method!
a)
\\(y = x + 3\\)
\\(3x + 2y = -4\\)

b)
\\(3a + b = 5\\)
\\(2a + 3b = -6\\)

1. mrs. hamilton is trying to plan a party for her math classes and receive two quotes. the hypotenuse hall charges \\$100 for a damage deposit and \\$6 per person for snacks. the pi place charges only \\$20 for a damage deposit but \\$10 per person for snacks. mrs. hamilton needs your help!

a) create a linear system of equations to model this situation. make sure to include statements that clearly identify any variables.

b) solve the system, clearly identifying your method.

Explanation:

Solve system 1a using substitution

We use substitution because the first equation is already solved for \(y\).

$$
LATEXBLOCK0
$$

Substitute Eq. 1 into Eq. 2:

$$
LATEXBLOCK1
$$

Substitute \(x = -2\) back into Eq. 1:

$$ y = -2 + 3 = 1 $$

The solution is \((x, y) = (-2, 1)\).

Solve system 1b using elimination

We use elimination to solve this system.

$$
LATEXBLOCK2
$$

Multiply Eq. 1 by \(-3\) to eliminate \(b\):

$$ -9a - 3b = -15 $$

Add this to Eq. 2:

$$
LATEXBLOCK3
$$

Substitute \(a = 3\) back into Eq. 1:

$$
LATEXBLOCK4
$$

The solution is \((a, b) = (3, -4)\).

Model the word problem in 2a

Let \(x\) represent the number of people attending.
Let \(C\) represent the total cost of the party.
For Hypotenuse Hall:

$$ C = 6x + 100 $$

For The Pi Place:

$$ C = 10x + 20 $$

The linear system is:

$$
LATEXBLOCK5
$$

Solve the word problem system in 2b

We use substitution since both equations are solved for \(C\).

$$
LATEXBLOCK6
$$

Substitute \(x = 20\) back to find \(C\):

$$ C = 6(20) + 100 = 120 + 100 = 220 $$

The solution is \(x = 20\) people and \(C = \$220\).

Answer:

Question 1a

• Method: Substitution Method
• Solution: \(x = -2\), \(y = 1\) (or as an ordered pair: \((-2, 1)\))

Question 1b

• Method: Elimination Method
• Solution: \(a = 3\), \(b = -4\) (or as an ordered pair: \((3, -4)\))

Question 2a

• Variables:
• Let \(x\) be the number of people attending the party.
• Let \(C\) be the total cost of the party in dollars.
• System of Equations:

$$
LATEXBLOCK0
$$

Question 2b

• Method: Substitution Method
• Solution: \(x = 20\) people, \(C = \$220\) (the costs are equal at \$220 for 20 people)