communicate
1. explain what an independent variable is.
2. explain what a dependent variable is.
3. describe how first differences of number sequences relate to linear equations.
4. describe the graph of a linear equation.
guided skills practice
applications
5. telecommunications the costs associated with a particular cellular phone plan are shown below. find the first differences and write an equation to represent the data pattern. (example 1)
hours per month 0 1 2 3 4 5
total cost ($) 25 37 49 61 73 85
6. transportation a hovercraft ferry crossing the english channel averages 47 miles per hour. write an equation, make a table, and draw a graph to show the distance the hovercraft travels for times from 0 to 3 hours. (example 2)
7. contests the prize for a baking contest is $200. if a contestant spends $6 per batch on ingredients to practice a new recipe, write an equation to describe the profit, p, the contestant will make if he wins in terms of the number of practice batches, b, used to perfect the recipe. (example 3)
practice and apply
find the first differences for each data set, and write an equation to represent the data pattern.
8. 0 1 2 3 4 5 6
17 23 29 35 41 47 53
9. 0 1 2 3 4 5 6
25 31 37 43 49 55 61
10. 0 1 2 3 4 5 6
60 180 300 420 540 660 780
11. 0 1 2 3 4 5 6
0 3 6 9 12 15 18

Question

Accepted Answer

### 1.
# Brief Explanations:
An independent variable is a variable whose value is not affected by other variables in a relationship. It is the input or the variable that is manipulated or chosen freely in an experiment or a mathematical - logical context.
# Answer:
An independent variable is a variable whose value is freely chosen or manipulated and is not determined by other variables in a relationship.

### 2.
# Brief Explanations:
A dependent variable is a variable whose value depends on the value of the independent variable. Its value changes as the independent variable changes.
# Answer:
A dependent variable is a variable whose value is determined by the value of an independent variable and changes in response to changes in the independent variable.

### 3.
# Brief Explanations:
In a linear equation of the form $y = mx + b$, if we have a number - sequence representing values of $y$ for consecutive integer values of $x$, the first differences (the differences between consecutive $y$ - values) are constant. This constant first - difference is equal to the slope $m$ of the linear equation.
# Answer:
For a number sequence related to a linear equation, the first differences are constant and equal to the slope of the linear equation.

### 4.
# Brief Explanations:
The graph of a linear equation $y=mx + b$ is a straight line. The slope $m$ determines the steepness and direction of the line, and the $y$ - intercept $b$ is the point where the line crosses the $y$ - axis.
# Answer:
The graph of a linear equation is a straight line, where the slope determines the steepness and direction, and the $y$ - intercept is the point of intersection with the $y$ - axis.

### 5.
# Explanation:
## Step1: Calculate first differences
For the cost data, when $x = 0$, $y = 25$; when $x = 1$, $y = 37$. The first difference between the cost values for $x = 1$ and $x = 0$ is $37-25=12$. Checking for other consecutive values: $49 - 37=12$, $61 - 49 = 12$, $73 - 61=12$, $85 - 73 = 12$.
## Step2: Find the equation
The general form of a linear equation is $y=mx + b$, where $m$ is the slope and $b$ is the $y$ - intercept. The slope $m$ is the first - difference, so $m = 12$. When $x = 0$, $y = 25$, so $b = 25$. The equation is $y=12x + 25$.
# Answer:
The first differences are 12. The equation is $y = 12x+25$

### 6.
# Explanation:
## Step1: Write the equation
The distance $d$ traveled by an object moving at a constant speed $v$ is given by the formula $d=vt$. Here, $v = 47$ miles per hour, so the equation is $d = 47t$.
## Step2: Make a table
| $t$ (hours) | $d$ (miles)|
|----|----|
|0|0|
|1|47|
|2|94|
|3|141|
## Step3: Draw a graph
The graph is a straight - line passing through the origin $(0,0)$ with a slope of 47. We plot the points from the table $(0,0)$, $(1,47)$, $(2,94)$, $(3,141)$ and draw a straight line through them.
# Answer:
Equation: $d = 47t$; Table: see above; Graph: a straight - line through $(0,0)$, $(1,47)$, $(2,94)$, $(3,141)$

### 7.
# Explanation:
## Step1: Write the profit equation
The prize is $200$ and the cost per batch is $6$. The profit $p$ in terms of the number of batches $b$ is given by the formula $p=200 - 6b$.
# Answer:
The equation is $p = 200-6b$

### 8.
# Explanation:
## Step1: Calculate first differences
$23-17 = 6$, $29 - 23=6$, $35 - 29 = 6$, $41 - 35=6$, $47 - 41 = 6$, $53 - 47=6$
## Step2: Find the equation
The general form of a linear equation is $y=mx + b$. The slope $m = 6$. When $x = 0$, $y = 17$, so $b = 17$. The equation is $y=6x + 17$.
# Answer:
The first differences are 6. The equation is $y = 6x+17$

### 9.
# Explanation:
## Step1: Calculate first differences
$31-25 = 6$, $37 - 31=6$, $43 - 37 = 6$, $49 - 43=6$, $55 - 49 = 6$, $61 - 55=6$
## Step2: Find the equation
The slope $m = 6$. When $x = 0$, $y = 25$, so $b = 25$. The equation is $y=6x + 25$.
# Answer:
The first differences are 6. The equation is $y = 6x+25$

### 10.
# Explanation:
## Step1: Calculate first differences
$180-60 = 120$, $300 - 180=120$, $420 - 300 = 120$, $540 - 420=120$, $660 - 540 = 120$, $780 - 660=120$
## Step2: Find the equation
The slope $m = 120$. When $x = 0$, $y = 60$, so $b = 60$. The equation is $y=120x + 60$.
# Answer:
The first differences are 120. The equation is $y = 120x+60$

### 11.
# Explanation:
## Step1: Calculate first differences
$3-0 = 3$, $6 - 3=3$, $9 - 6 = 3$, $12 - 9=3$, $15 - 12 = 3$, $18 - 15=3$
## Step2: Find the equation
The slope $m = 3$. When $x = 0$, $y = 0$, so $b = 0$. The equation is $y=3x$.
# Answer:
The first differences are 3. The equation is $y = 3x$

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