Theresa and Tim read books. Theresa has read 2 books and reads 6 more each week. Tim has read 4 books and reads 6 more each week. After how many weeks will they have read the same number of books?

Question

Write an equation that represents the number of books Tim has read, y, after x weeks.
Great job!
Complete the table below with the number of books, y, that Tim has read after 2, 6, and 10 weeks. You can solve on paper and then enter your answer on Zearn.
y=4+6x

Weeks (x) Books (y)
2
6
10
Great job!
Next, graph the points you just found on the coordinate plane to show the relationship between x, the number of weeks, and y, the number of books Tim has read.
Weeks (x) Books (y)
2 16
6 40
10 64
Good work!
Now, consider Theresa. Write an equation that represents the number of comic books Theresa has read, y, after x weeks.
You got it!
Complete the table below with the number of books, y, that Theresa has read after various numbers of weeks, x. You can solve on paper and then enter your answer on Zearn.
y=2+6x

Weeks (x) Books (y)
1
5
9
Excellent!
The line represents the equation you graphed earlier, showing the number of books Tim has read, y, after x weeks. Use the points that you just found to graph the equation for Theresa on this same graph.
Weeks (x) Books (y)
1 8
5 32
9 56
You got it!
Is there a time when Theresa and Tim have read the same number of books?
x
y
Th
Ti
0
1
2
3
4
5
6
7
8
9
10
0
8
16
24
32
40
48
56
64
72
80
Weeks
Books read
a time when they have read the same number of books.

We can see this on the graph

Answer 1

To determine when Theresa and Tim will have read the same number of books, we need to set their equations equal to each other and solve for \( x \).

Theresa's equation:
\[ y = 2 + 6x \]

Tim's equation:
\[ y = 4 + 6x \]

Setting them equal to each other: \[ 2 + 6x = 4 + 6x \]

Now, we can simplify this by subtracting \( 6x \) from both sides: \[ 2 = 4 \]

This tells us that there is no solution to the equation, meaning Tim and Theresa will never read the same number of books after the starting conditions given (where both have a specific number of books read and the same weekly reading rate).

From the table values calculated, we already confirmed that after 2, 6, and 10 weeks, Tim will reach 16, 40, and 64 books respectively, while Theresa's books are given as:

• After 1 week: \( y = 2 + 6(1) = 8 \)
• After 5 weeks: \( y = 2 + 6(5) = 32 \)
• After 9 weeks: \( y = 2 + 6(9) = 56 \)

This confirms when plotted, the graphs for Theresa and Tim will never intersect, indicating that there will never be a time when they will have read the same number of books under these specific conditions.

So, the answer is: No, there is no time when Theresa and Tim have read the same number of books.