Question

1. decide whether each of the following is linear or not linear. in each case, explain your answer. (1 point each)

a. 3x + 4y+2 = 0
b.

x	y
2	1
4	5
6	15
8	30

c. you have a secret that you tell to one person. every hour, each of the people that know the secret tells one person. the number of people who know is n, and t is the number of hours since you told the first person. is n a linear function of t?

Explanation:

Step1: Recall linear - equation form

A linear equation is of the form $ax + by+ c = 0$, where $a$, $b$, and $c$ are constants and $x$ and $y$ are variables, and the exponents of $x$ and $y$ are 1.
For the equation $3x + 4y+2 = 0$, $a = 3$, $b = 4$, $c = 2$. The exponents of $x$ and $y$ are 1. So it is linear.

Step2: Check for linearity of data - set

For a set of data points $(x,y)$ to represent a linear relationship, the slope between any two points should be the same. The slope $m=\frac{y_2 - y_1}{x_2 - x_1}$.
For the points $(2,1)$ and $(4,5)$: $m_1=\frac{5 - 1}{4 - 2}=\frac{4}{2}=2$.
For the points $(4,5)$ and $(6,15)$: $m_2=\frac{15 - 5}{6 - 4}=\frac{10}{2}=5$.
Since $m_1
eq m_2$, the relationship is not linear.

Step3: Analyze the secret - spreading situation

Initially, 1 person knows the secret ($t = 0$, $N = 1$). After 1 hour ($t = 1$), 2 people know the secret. After 2 hours ($t = 2$), 4 people know the secret. The relationship is $N = 2^t$, which is an exponential function, not a linear function.

Answer:

a. Linear. It is in the form $ax+by + c=0$ with exponents of $x$ and $y$ equal to 1.
b. Not linear. The slopes between different pairs of points are not equal.
c. Not linear. The relationship is an exponential function $N = 2^t$.