In the given problem, we need to determine which of the ordered pairs could appear in the table if the relationship between days and books collected is linear. Let's analyze the linear relationship based on the points provided in the table: (1, 18), (3, 28), and (5, 38).
First, we'll calculate the slope ([tex]\( m \)[/tex]) of the linear relationship. The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the points (1, 18) and (3, 28):
[tex]\[ m = \frac{28 - 18}{3 - 1} = \frac{10}{2} = 5 \][/tex]
So the slope ([tex]\( m \)[/tex]) is 5.
Next, we find the y-intercept ([tex]\( b \)[/tex]) using the equation of the line [tex]\( y = mx + b \)[/tex]. We can use one of the given points (let's use (1, 18)) to solve for [tex]\( b \)[/tex]:
[tex]\[ 18 = 5 \cdot 1 + b \][/tex]
[tex]\[ 18 = 5 + b \][/tex]
[tex]\[ b = 18 - 5 = 13 \][/tex]
So, the equation of the line is:
[tex]\[ y = 5x + 13 \][/tex]
Now, we'll check each pair to see if they satisfy this equation:
1. For [tex]\((0, 8)\)[/tex]:
[tex]\[ y = 5 \cdot 0 + 13 = 13 \][/tex]
This does not match [tex]\( y = 8 \)[/tex], so [tex]\((0, 8)\)[/tex] does not appear in the table.
2. For [tex]\( (2, 23) \)[/tex]:
[tex]\[ y = 5 \cdot 2 + 13 = 10 + 13 = 23 \][/tex]
This matches [tex]\( y = 23 \)[/tex], so [tex]\( (2, 23) \)[/tex] could appear in the table.
3. For [tex]\( (4, 32) \)[/tex]:
[tex]\[ y = 5 \cdot 4 + 13 = 20 + 13 = 33 \][/tex]
This does not match [tex]\( y = 32 \)[/tex], so [tex]\( (4, 32) \)[/tex] does not appear in the table.
4. For [tex]\( (6, 48) \)[/tex]:
[tex]\[ y = 5 \cdot 6 + 13 = 30 + 13 = 43 \][/tex]
This does not match [tex]\( y = 48 \)[/tex], so [tex]\( (6, 48) \)[/tex] does not appear in the table.
5. For [tex]\( (7, 48) \)[/tex]:
[tex]\[ y = 5 \cdot 7 + 13 = 35 + 13 = 48 \][/tex]
This matches [tex]\( y = 48 \)[/tex], so [tex]\( (7, 48) \)[/tex] could appear in the table.
Based on our analysis, the ordered pairs that could appear in the table are:
[tex]\[ (2, 23) \][/tex]
[tex]\[ (7, 48) \][/tex]
