Answer :
To determine the equations that model the total yards of fabric Sophie will buy, we need to check which equations are satisfied by all the given combinations of [tex]\( x \)[/tex] (red fabric in yards) and [tex]\( y \)[/tex] (blue fabric in yards) provided in the table.
The given combinations are:
[tex]\[ \begin{array}{|c|c|} \hline \text{Red fabric } (x) & \text{Blue fabric } (y) \\ \hline 1 & 27 \\ \hline 2 & 26 \\ \hline 3 & 25 \\ \hline 4 & 24 \\ \hline \end{array} \][/tex]
We will check each equation:
1. Equation: [tex]\( x + y = 28 \)[/tex]
- For [tex]\( x = 1 \)[/tex] and [tex]\( y = 27 \)[/tex]:
[tex]\[ 1 + 27 = 28 \][/tex]
- For [tex]\( x = 2 \)[/tex] and [tex]\( y = 26 \)[/tex]:
[tex]\[ 2 + 26 = 28 \][/tex]
- For [tex]\( x = 3 \)[/tex] and [tex]\( y = 25 \)[/tex]:
[tex]\[ 3 + 25 = 28 \][/tex]
- For [tex]\( x = 4 \)[/tex] and [tex]\( y = 24 \)[/tex]:
[tex]\[ 4 + 24 = 28 \][/tex]
This equation [tex]\( x + y = 28 \)[/tex] is satisfied by all the given pairs.
2. Equation: [tex]\( 28 + x = y \)[/tex]
- For [tex]\( x = 1 \)[/tex] and [tex]\( y = 27 \)[/tex]:
[tex]\[ 28 + 1 = 29 \ (\text{which is not } 27) \][/tex]
This equation [tex]\( 28 + x = y \)[/tex] is not satisfied by the pairs, so it is invalid.
3. Equation: [tex]\( x - y = 28 \)[/tex]
- For [tex]\( x = 1 \)[/tex] and [tex]\( y = 27 \)[/tex]:
[tex]\[ 1 - 27 = -26 \ (\text{which is not } 28) \][/tex]
This equation [tex]\( x - y = 28 \)[/tex] is not satisfied by the pairs, so it is invalid.
4. Equation: [tex]\( 28 - x = y \)[/tex]
- For [tex]\( x = 1 \)[/tex] and [tex]\( y = 27 \)[/tex]:
[tex]\[ 28 - 1 = 27 \][/tex]
- For [tex]\( x = 2 \)[/tex] and [tex]\( y = 26 \)[/tex]:
[tex]\[ 28 - 2 = 26 \][/tex]
- For [tex]\( x = 3 \)[/tex] and [tex]\( y = 25 \)[/tex]:
[tex]\[ 28 - 3 = 25 \][/tex]
- For [tex]\( x = 4 \)[/tex] and [tex]\( y = 24 \)[/tex]:
[tex]\[ 28 - 4 = 24 \][/tex]
This equation [tex]\( 28 - x = y \)[/tex] is satisfied by all the given pairs.
5. Equation: [tex]\( 28 - y = x \)[/tex]
- For [tex]\( x = 1 \)[/tex] and [tex]\( y = 27 \)[/tex]:
[tex]\[ 28 - 27 = 1 \][/tex]
- For [tex]\( x = 2 \)[/tex] and [tex]\( y = 26 \)[/tex]:
[tex]\[ 28 - 26 = 2 \][/tex]
- For [tex]\( x = 3 \)[/tex] and [tex]\( y = 25 \)[/tex]:
[tex]\[ 28 - 25 = 3 \][/tex]
- For [tex]\( x = 4 \)[/tex] and [tex]\( y = 24 \)[/tex]:
[tex]\[ 28 - 24 = 4 \][/tex]
This equation [tex]\( 28 - y = x \)[/tex] is satisfied by all the given pairs.
Therefore, the valid equations that model the total yards of fabric Sophie will buy are:
- [tex]\( x + y = 28 \)[/tex]
- [tex]\( 28 - x = y \)[/tex]
- [tex]\( 28 - y = x \)[/tex]
Thus, the correct checked equations are:
[tex]\[ x + y = 28 \][/tex]
[tex]\[ 28 - x = y \][/tex]
[tex]\[ 28 - y = x \][/tex]
The given combinations are:
[tex]\[ \begin{array}{|c|c|} \hline \text{Red fabric } (x) & \text{Blue fabric } (y) \\ \hline 1 & 27 \\ \hline 2 & 26 \\ \hline 3 & 25 \\ \hline 4 & 24 \\ \hline \end{array} \][/tex]
We will check each equation:
1. Equation: [tex]\( x + y = 28 \)[/tex]
- For [tex]\( x = 1 \)[/tex] and [tex]\( y = 27 \)[/tex]:
[tex]\[ 1 + 27 = 28 \][/tex]
- For [tex]\( x = 2 \)[/tex] and [tex]\( y = 26 \)[/tex]:
[tex]\[ 2 + 26 = 28 \][/tex]
- For [tex]\( x = 3 \)[/tex] and [tex]\( y = 25 \)[/tex]:
[tex]\[ 3 + 25 = 28 \][/tex]
- For [tex]\( x = 4 \)[/tex] and [tex]\( y = 24 \)[/tex]:
[tex]\[ 4 + 24 = 28 \][/tex]
This equation [tex]\( x + y = 28 \)[/tex] is satisfied by all the given pairs.
2. Equation: [tex]\( 28 + x = y \)[/tex]
- For [tex]\( x = 1 \)[/tex] and [tex]\( y = 27 \)[/tex]:
[tex]\[ 28 + 1 = 29 \ (\text{which is not } 27) \][/tex]
This equation [tex]\( 28 + x = y \)[/tex] is not satisfied by the pairs, so it is invalid.
3. Equation: [tex]\( x - y = 28 \)[/tex]
- For [tex]\( x = 1 \)[/tex] and [tex]\( y = 27 \)[/tex]:
[tex]\[ 1 - 27 = -26 \ (\text{which is not } 28) \][/tex]
This equation [tex]\( x - y = 28 \)[/tex] is not satisfied by the pairs, so it is invalid.
4. Equation: [tex]\( 28 - x = y \)[/tex]
- For [tex]\( x = 1 \)[/tex] and [tex]\( y = 27 \)[/tex]:
[tex]\[ 28 - 1 = 27 \][/tex]
- For [tex]\( x = 2 \)[/tex] and [tex]\( y = 26 \)[/tex]:
[tex]\[ 28 - 2 = 26 \][/tex]
- For [tex]\( x = 3 \)[/tex] and [tex]\( y = 25 \)[/tex]:
[tex]\[ 28 - 3 = 25 \][/tex]
- For [tex]\( x = 4 \)[/tex] and [tex]\( y = 24 \)[/tex]:
[tex]\[ 28 - 4 = 24 \][/tex]
This equation [tex]\( 28 - x = y \)[/tex] is satisfied by all the given pairs.
5. Equation: [tex]\( 28 - y = x \)[/tex]
- For [tex]\( x = 1 \)[/tex] and [tex]\( y = 27 \)[/tex]:
[tex]\[ 28 - 27 = 1 \][/tex]
- For [tex]\( x = 2 \)[/tex] and [tex]\( y = 26 \)[/tex]:
[tex]\[ 28 - 26 = 2 \][/tex]
- For [tex]\( x = 3 \)[/tex] and [tex]\( y = 25 \)[/tex]:
[tex]\[ 28 - 25 = 3 \][/tex]
- For [tex]\( x = 4 \)[/tex] and [tex]\( y = 24 \)[/tex]:
[tex]\[ 28 - 24 = 4 \][/tex]
This equation [tex]\( 28 - y = x \)[/tex] is satisfied by all the given pairs.
Therefore, the valid equations that model the total yards of fabric Sophie will buy are:
- [tex]\( x + y = 28 \)[/tex]
- [tex]\( 28 - x = y \)[/tex]
- [tex]\( 28 - y = x \)[/tex]
Thus, the correct checked equations are:
[tex]\[ x + y = 28 \][/tex]
[tex]\[ 28 - x = y \][/tex]
[tex]\[ 28 - y = x \][/tex]
