In order to find the intercepts, substitute 0 for one of the variables and solve the equation for the other variable.
B
b
The product of odd numbers is always odd. The difference between an even number and an odd number is always odd.
C
c
Plot the intercepts and connect them with a line segment. Remember that the number of baskets cannot be negative.
A
a
x-intercept: (0,27)
y-intercept: (18,0) Interpretation: See solution.
B
b
No, see solution.
C
c
Possible Solutions: (9,12) and (18,6)
Practice makes perfect
a
The given equation models the total score of a team that makes only 2-point and 3-point baskets. In the equation, x is the number of 2-point baskets made and y is the number of 3-point baskets made.
2x+3y=54We will find the intercepts starting with the x-intercept. In order to find the x-intercept, we substitute 0 for y and solve the equation for x.
The point (27,0) is the x-intercept, which means that the team did not make any 3-point baskets if it made 27 2-point baskets. Next, we will find the y-intercept. To do so, we will substitute 0 for x and solve for y.
The y-intercept of the equation is the point (0,18). The y-intercept tells us if the team made 18 3-point baskets, then they did not make any 2-point baskets.
b
In order to determine whether the number of 3-point baskets can be odd, we will consider two statements.
The product of odd numbers is always odd.
The difference between an even number and an odd number is always odd.
Considering these two statements, if the number of 3-point baskets is odd, then the score made by making 2-point baskets would also have to be odd. Remember that an odd number is an integer that is not even. Let's first recall the definition of an even number.
Even number
An even number is any integer that is divisible exactly by 2.
In other words, an odd number is not wholly divisible by 2, meaning that no odd amount of points can be scored by making 2-baskets. Therefore, the number of 3-point baskets cannot be an odd number.
c
In order to graph the equation, we will use the intercepts that we found in Part A. We will plot them on the coordinate plane and connect them with a line segment. Notice that the number of the baskets cannot be negative.
Only whole-number values of x and y make sense in the context of the exercise. Other than the intercepts, we can see that the line passes through the points (9,12) and (18,6). Therefore, only these two additional points could be solutions to the equation.
