Field goals are worth 3 points and touchdowns are worth 7 points in football. A team scored a total of 63 points.
1) Determine the variables.
2) Determine the x-intercept and the y-intercept.
3) If they scored 3 touchdowns, how many field goals did they score?
4) Find the domain and range.
x = FGs
y = TDs
3x+7y = 63
The intercept form is
x/21 + y/9 = 1
and so on
1) The variables in this scenario are the number of field goals (F) and the number of touchdowns (T).
2) To determine the x-intercept, we need to find the number of field goals (F) when the total points (Y) is 0. Since field goals are worth 3 points and touchdowns are worth 7 points, we can set up the equation:
3F + 7T = 0
Solving this equation for F, we get F = 0. This means that when the total points are 0, the team did not score any field goals. Therefore, the x-intercept is (0,0).
To determine the y-intercept, we need to find the number of touchdowns (T) when the total points (Y) is 0. Using the same equation as above, we have:
3F + 7T = 0
Solving this equation for T, we get T = 0. This means that when the total points are 0, the team did not score any touchdowns. Therefore, the y-intercept is (0,0).
3) If the team scored 3 touchdowns, we can substitute T = 3 into the equation and find the value of F. Using the equation:
3F + 7T = 63
Substituting T = 3, we get:
3F + 7(3) = 63
Simplifying, we have:
3F + 21 = 63
Subtracting 21 from both sides, we get:
3F = 42
Dividing both sides by 3, we find:
F = 14
Therefore, if the team scored 3 touchdowns, they scored 14 field goals.
4) The domain in this scenario represents the possible values for the number of field goals (F) and touchdowns (T). Since the team cannot score a negative number of field goals or touchdowns, the domain for both F and T is the set of non-negative integers (0, 1, 2, 3, ...).
The range represents the possible values for the total points (Y). Since the team scored a total of 63 points, the range for Y is {63}.
1) The variables in this scenario would be the number of field goals and the number of touchdowns the team scored.
Let's call the number of field goals "F" and the number of touchdowns "T".
2) To determine the x-intercept, we need to find the point where the team scores 0 points. Since touchdowns are worth 7 points and field goals are worth 3 points, we can set up the equation:
7T + 3F = 0
Solving this equation, we find that the x-intercept occurs when both the number of touchdowns and field goals scored are 0.
To determine the y-intercept, we need to find the point where the team scores 63 points. So we set up the equation:
7T + 3F = 63
Solving this equation, we can find the number of touchdowns and field goals that would result in the team scoring 63 points.
3) If they scored 3 touchdowns, we can substitute T = 3 into the equation we set up in step 2:
7(3) + 3F = 63
Simplifying the equation, we have:
21 + 3F = 63
Subtracting 21 from both sides, we get:
3F = 42
Dividing both sides by 3, we find:
F = 14
So, if the team scored 3 touchdowns, they would also have scored 14 field goals.
4) The domain in this scenario would be the possible values for the number of touchdowns and field goals. Since both values must be non-negative integers (we can't score negative touchdowns or field goals), the domain is the set of non-negative integers for both T and F.
The range represents the possible values for the team's score, which in this case is 63 points. So the range would be the set {63}.