6 from unit 2, lesson 17
elena writes the equation 6x + 2y = 12. write a new equation that has:
a. exactly one solution in common with elena’s equation
b. no solutions in common with elena’s equation
c. infinitely many solutions in common with elena’s equation
7 from unit 3, lesson 9
a restaurant owner wants to see if there is a relationship between the amount of sugar in some food items on her menu and how popular the items are.
she creates a scatter plot to show the relationship between amount of sugar in menu items and the number of orders for those items. the correlation coefficient for the line of best fit is 0.58.
a. are the two variables correlated?
explain your reasoning.
b. is it likely or unlikely that one of the variables causes the other to change?
explain your reasoning.

Question

Accepted Answer

### Problem 6
#### Part a
# Explanation:
## Step1: Recall solution conditions
For a system of linear equations $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$, exactly one solution occurs when $\frac{a_1}{a_2}
eq\frac{b_1}{b_2}$. Elena's equation is $6x + 2y = 12$. Let's choose a new equation, say $x + y = 1$ (here $a_1 = 6, b_1 = 2$; $a_2 = 1, b_2 = 1$, and $\frac{6}{1}
eq\frac{2}{1}$).
## Step2: Verify
The system $\begin{cases}6x + 2y = 12\x + y = 1\end{cases}$ can be solved. From the second equation, $y = 1 - x$. Substitute into the first: $6x + 2(1 - x)=12\Rightarrow6x + 2 - 2x = 12\Rightarrow4x = 10\Rightarrow x=\frac{5}{2}$, $y = 1-\frac{5}{2}=-\frac{3}{2}$, so only one solution.
# Answer: $x + y = 1$ (any equation with $\frac{6}{a_2}
eq\frac{2}{b_2}$ works)

#### Part b
# Explanation:
## Step1: Recall no - solution condition
For a system of linear equations $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$, no solutions occur when $\frac{a_1}{a_2}=\frac{b_1}{b_2}
eq\frac{c_1}{c_2}$. Elena's equation is $6x + 2y = 12$. Let's take $6x + 2y = 1$. Here $\frac{6}{6}=\frac{2}{2}=1$, but $\frac{12}{1}=12
eq1$.
## Step2: Verify
The system $\begin{cases}6x + 2y = 12\6x + 2y = 1\end{cases}$ has the same left - hand side but different right - hand sides, so no solutions.
# Answer: $6x + 2y = 1$ (any equation with $\frac{6}{a_2}=\frac{2}{b_2}
eq\frac{12}{c_2}$ works)

#### Part c
# Explanation:
## Step1: Recall infinite - solutions condition
For a system of linear equations $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$, infinite solutions occur when $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$. Elena's equation is $6x + 2y = 12$. Let's multiply Elena's equation by $\frac{1}{2}$, we get $3x + y = 6$. Here $\frac{6}{3}=\frac{2}{1}=\frac{12}{6}=2$.
## Step2: Verify
The equation $3x + y = 6$ can be rewritten as $6x + 2y = 12$ (multiply both sides by 2), so every solution of Elena's equation is a solution of $3x + y = 6$ and vice - versa.
# Answer: $3x + y = 6$ (any equation that is a non - zero multiple of $6x + 2y = 12$ works)

### Problem 7
#### Part a
# Explanation:
## Step1: Recall correlation coefficient meaning
The correlation coefficient $r$ ranges from $- 1$ to $1$. If $|r|>0$, there is a correlation. A positive $r$ indicates a positive correlation, a negative $r$ indicates a negative correlation. The closer $|r|$ is to $1$, the stronger the correlation.
## Step2: Analyze the given coefficient
The correlation coefficient is $r = 0.58$. Since $|0.58|=0.58>0$, there is a positive correlation between the amount of sugar in menu items and the number of orders for those items.
# Answer: Yes, the two variables are correlated. The correlation coefficient is $0.58$, and since $|0.58|>0$, there is a positive correlation between the amount of sugar in menu items and the number of orders for those items.

#### Part b
# Explanation:
## Step1: Recall correlation vs causation
Correlation does not imply causation. Just because two variables are correlated (in this case, the amount of sugar and the number of orders) does not mean that one variable causes the other to change. There could be other factors (e.g., the taste of the item, the price, the time of day it is served, etc.) that influence the number of orders.
## Step2: Apply to the problem
Even though there is a positive correlation between the amount of sugar and the number of orders, it is unlikely that the amount of sugar alone causes the change in the number of orders. For example, a menu item with a lot of sugar might also be very flavorful, and it's the flavor (or other factors) that causes people to order it more, not just the sugar content.
# Answer: It is unlikely that one of the variables causes the other to change. Correlation (a value of $0.58$ here) does not imply causation. There could be other confounding factors (such as taste, price, or marketing of the menu items) that affect the number of orders, rather than the amount of sugar alone causing the change in the number of orders.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 6

Part a

Step1: Recall solution conditions

Step2: Verify

Step1: Recall no - solution condition

Step2: Verify

Step1: Recall infinite - solutions condition

Step2: Verify

Answer:

Part b