Answer :
Let's solve this step-by-step to determine which equation will help Tisha find how many points she needs to earn in the district match to advance to the state finals, given the total points required [tex]\( T \)[/tex], points earned in three local matches [tex]\( a \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex], and district match points [tex]\( b \)[/tex].
### Understanding the Problem
1. Total Points Required ([tex]\( T \)[/tex]): Tisha's team needs [tex]\( T \)[/tex] points to advance.
2. Local Matches: They have earned points in three local matches, denoted as [tex]\( a \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex].
3. District Match Points ([tex]\( b \)[/tex]): Points to be earned in the district match.
### Key Information
- Points from the district match are worth three times the points from local matches.
- The goal is to find out how many points [tex]\( b \)[/tex] need to be earned in the district match given the points from the local matches and the total points required [tex]\( T \)[/tex].
### Developing the Equation
We need an equation that relates the total points required [tex]\( T \)[/tex], the local match points [tex]\( a \)[/tex], [tex]\( c \)[/tex], [tex]\( d \)[/tex], and the district match points [tex]\( b \)[/tex].
### Assessment of Each Given Equation
1. Equation 1: [tex]\( b = \frac{T}{3} - a - c - d \)[/tex]
- This implies that the points required in the district match [tex]\( b \)[/tex] are equal to the total points divided by 3 minus the sum of the local match points [tex]\( a \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex].
2. Equation 2: [tex]\( b = \frac{T + a + c + d}{3} \)[/tex]
- This suggests that the district points [tex]\( b \)[/tex] are equal to the sum of the total points required [tex]\( T \)[/tex], and the local match points divided by 3.
3. Equation 3: [tex]\( b = \frac{T - a - c - d}{3} \)[/tex]
- This indicates that the district match points [tex]\( b \)[/tex] are equal to the total points required minus the sum of the local points, divided by 3.
4. Equation 4: [tex]\( b = 3(T - a - b - d) \)[/tex]
- This equation suggests that the district match points [tex]\( b \)[/tex] are three times the difference between the total points, the local matches non-district points.
### Verifying the Correct Equation
1. To advance, the total points from the local matches ([tex]\( a \)[/tex], [tex]\( c \)[/tex], [tex]\( d \)[/tex]) and the district match points multiplied by 3 (since district points are thrice the value) should equal the required total points [tex]\( T \)[/tex].
2. The correct way to calculate [tex]\( b \)[/tex] should account for the fact that district match points are worth three times the local points, thus we adjust the equation accordingly.
Let's break it down:
- Let [tex]\( b \)[/tex] be the points from the district match.
- Since district match points are worth three times the local match points, the contribution of [tex]\( b \)[/tex] to the total [tex]\( T \)[/tex] would be [tex]\( 3b \)[/tex] approximately.
- Hence, the summation of points should be:
[tex]\[ a + c + d + 3b = T \][/tex]
Rearranging to solve for [tex]\( b \)[/tex]:
[tex]\[ 3b = T - (a + c + d) \][/tex]
[tex]\[ b = \frac{T - a - c - d}{3} \][/tex]
### Conclusion
The correct equation to determine how many points Tisha needs to earn in the district match is:
[tex]\[ b = \frac{T - a - c - d}{3} \][/tex]
Thus, the correct choice for the equation is:
[tex]\[ b = \frac{T - a - c - d}{3} \][/tex]
### Understanding the Problem
1. Total Points Required ([tex]\( T \)[/tex]): Tisha's team needs [tex]\( T \)[/tex] points to advance.
2. Local Matches: They have earned points in three local matches, denoted as [tex]\( a \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex].
3. District Match Points ([tex]\( b \)[/tex]): Points to be earned in the district match.
### Key Information
- Points from the district match are worth three times the points from local matches.
- The goal is to find out how many points [tex]\( b \)[/tex] need to be earned in the district match given the points from the local matches and the total points required [tex]\( T \)[/tex].
### Developing the Equation
We need an equation that relates the total points required [tex]\( T \)[/tex], the local match points [tex]\( a \)[/tex], [tex]\( c \)[/tex], [tex]\( d \)[/tex], and the district match points [tex]\( b \)[/tex].
### Assessment of Each Given Equation
1. Equation 1: [tex]\( b = \frac{T}{3} - a - c - d \)[/tex]
- This implies that the points required in the district match [tex]\( b \)[/tex] are equal to the total points divided by 3 minus the sum of the local match points [tex]\( a \)[/tex], [tex]\( c \)[/tex], and [tex]\( d \)[/tex].
2. Equation 2: [tex]\( b = \frac{T + a + c + d}{3} \)[/tex]
- This suggests that the district points [tex]\( b \)[/tex] are equal to the sum of the total points required [tex]\( T \)[/tex], and the local match points divided by 3.
3. Equation 3: [tex]\( b = \frac{T - a - c - d}{3} \)[/tex]
- This indicates that the district match points [tex]\( b \)[/tex] are equal to the total points required minus the sum of the local points, divided by 3.
4. Equation 4: [tex]\( b = 3(T - a - b - d) \)[/tex]
- This equation suggests that the district match points [tex]\( b \)[/tex] are three times the difference between the total points, the local matches non-district points.
### Verifying the Correct Equation
1. To advance, the total points from the local matches ([tex]\( a \)[/tex], [tex]\( c \)[/tex], [tex]\( d \)[/tex]) and the district match points multiplied by 3 (since district points are thrice the value) should equal the required total points [tex]\( T \)[/tex].
2. The correct way to calculate [tex]\( b \)[/tex] should account for the fact that district match points are worth three times the local points, thus we adjust the equation accordingly.
Let's break it down:
- Let [tex]\( b \)[/tex] be the points from the district match.
- Since district match points are worth three times the local match points, the contribution of [tex]\( b \)[/tex] to the total [tex]\( T \)[/tex] would be [tex]\( 3b \)[/tex] approximately.
- Hence, the summation of points should be:
[tex]\[ a + c + d + 3b = T \][/tex]
Rearranging to solve for [tex]\( b \)[/tex]:
[tex]\[ 3b = T - (a + c + d) \][/tex]
[tex]\[ b = \frac{T - a - c - d}{3} \][/tex]
### Conclusion
The correct equation to determine how many points Tisha needs to earn in the district match is:
[tex]\[ b = \frac{T - a - c - d}{3} \][/tex]
Thus, the correct choice for the equation is:
[tex]\[ b = \frac{T - a - c - d}{3} \][/tex]
