Final answer:
The equation that represents the linear relationship between the number of weeks of practice and the total number of juggling tricks learned is J = 7W - 5.
Step-by-step explanation:
The student has noticed a linear relationship between the number of weeks of practice and the total number of juggling tricks learned. To find the equation that represents this relationship, we can use two points from the given information: week 1 (W = 1, J = 2) and week 2 (W = 2, J = 9).
First, we can find the slope (m) using the formula:
m = (J2 - J1) / (W2 - W1)
Plugging in the values, we get:
m = (9 - 2) / (2 - 1)
Simplifying, we get:
m = 7 / 1 = 7
Next, we can use the point-slope form of a linear equation:
Y - Y1 = m(X - X1)
Plugging in the values of one of the points, let's use (W = 1, J = 2), we get:
y - 2 = 7(x - 1)
Simplifying, we get the equation:
J = 7W - 5
Trevor observed a linear relationship in the total number of juggling tricks (J) he learned over the weeks of practice (W). To express this relationship mathematically, he wrote the equation
�
=
7
�
−
5
J=7W−5. The equation is derived from the pattern he observed: each week, he added 7 new tricks, starting from 2 in the first week. The constant term of -5 accounts for the initial number of tricks he knew before he started practicing juggling. In week 1 (
�
=
1
W=1), the equation evaluates to
�
=
7
(
1
)
−
5
=
2
J=7(1)−5=2, matching the initial number of tricks. In week 2 (
�
=
2
W=2), it becomes
�
=
7
(
2
)
−
5
=
9
J=7(2)−5=9, and in week 3 (
�
=
3
W=3), it becomes
�
=
7
(
3
)
−
5
=
16
J=7(3)−5=16, consistent with the observed values. Thus,
�
=
7
�
−
5
J=7W−5 represents Trevor's linear relationship between weeks of practice and the total number of juggling tricks learned.