Question

column a
column b
final answer

1. find the value of k so that the system has infinitely many solutions

$\

$$\begin{cases} y = 10 - 4x \\\\ 2y = 20 + kx \\end{cases}$$

$

1. find the value of k so that the system has no solutions.

$\

$$\begin{cases} y = -20 - 8x \\\\ y = 13 + kx \\end{cases}$$

$

1. a pile of coins consists of quarters and dimes. the total value of the pile is $2.05. there are 4 more quarters than dimes. find the number of quarters in the pile.
2. raquel works at the carwash earning $11 an hour and baby-sits for her neighbor for $8 an hour. one week she worked a combined total of 13 hours and earned $125. how many hours did she work at the carwash?
3. the graph of a system of inequalities is shown below. which point represents a solution to the system?
4. the graph of a system of inequalities is shown below. which point represents an ordered pair that is a solution to neither inequality?

Explanation:

---

Column A, Problem 4

Step1: Rewrite first equation

Multiply first equation by 2:
$$2y = 20 - 8x$$

Step2: Match coefficients for infinite solutions

For infinite solutions, coefficients of $x$ must be equal:
$$k = -8$$

---

Column B, Problem 4

Step1: Identify parallel line condition

For no solutions, slopes are equal:
$$k = -8$$

---

Column A, Problem 5

Step1: Define variables

Let $q$ = quarters, $d$ = dimes. $q = d + 4$

Step2: Set up value equation

$$0.25q + 0.10d = 2.05$$

Step3: Substitute $q = d+4$

$$0.25(d+4) + 0.10d = 2.05$$
$$0.25d + 1 + 0.10d = 2.05$$
$$0.35d = 1.05$$

Step4: Solve for $d$, then $q$

$$d = \frac{1.05}{0.35} = 3$$
$$q = 3 + 4 = 7$$

---

Column B, Problem 5

Step1: Define variables

Let $c$ = carwash hours, $b$ = babysit hours. $c + b = 13$

Step2: Set up earnings equation

$$11c + 8b = 125$$

Step3: Substitute $b = 13 - c$

$$11c + 8(13 - c) = 125$$
$$11c + 104 - 8c = 125$$
$$3c = 21$$

Step4: Solve for $c$

$$c = \frac{21}{3} = 7$$

---

Column A, Problem 6

Step1: Identify solution region

A solution lies in the overlapping shaded area. Point B is in the intersection of both shaded regions.

---

Column B, Problem 6

Step1: Identify non-solution region

A point not in either shaded area is C (it lies outside both inequality regions).

---

Answer:

Column A 4: $\boldsymbol{k=-8}$
Column B 4: $\boldsymbol{k=-8}$
Column A 5: $\boldsymbol{7}$ quarters
Column B 5: $\boldsymbol{7}$ hours
Column A 6: $\boldsymbol{\text{Point B}}$
Column B 6: $\boldsymbol{\text{Point C}}$