To determine which equations model the total yards of fabric Sophie will buy, we need to evaluate each given equation using the pairs of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] from the table. Let’s examine each equation one by one with the provided combinations of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
Here are the pairs given:
- [tex]\( (x, y) = (1, 27) \)[/tex]
- [tex]\( (x, y) = (2, 26) \)[/tex]
- [tex]\( (x, y) = (3, 25) \)[/tex]
- [tex]\( (x, y) = (4, 24) \)[/tex]
1. [tex]\( x + y = 28 \)[/tex]
Check if the sum of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] equals 28 for each pair:
- For [tex]\( (1, 27) \)[/tex]: [tex]\( 1 + 27 = 28 \)[/tex]
- For [tex]\( (2, 26) \)[/tex]: [tex]\( 2 + 26 = 28 \)[/tex]
- For [tex]\( (3, 25) \)[/tex]: [tex]\( 3 + 25 = 28 \)[/tex]
- For [tex]\( (4, 24) \)[/tex]: [tex]\( 4 + 24 = 28 \)[/tex]
All pairs satisfy the equation [tex]\( x + y = 28 \)[/tex].
2. [tex]\( 28 + x = y \)[/tex]
Check if adding 28 to [tex]\(x\)[/tex] gives [tex]\(y\)[/tex] for each pair:
- For [tex]\( (1, 27) \)[/tex]: [tex]\( 28 + 1 = 29 \neq 27 \)[/tex]
- For [tex]\( (2, 26) \)[/tex]: [tex]\( 28 + 2 = 30 \neq 26 \)[/tex]
- For [tex]\( (3, 25) \)[/tex]: [tex]\( 28 + 3 = 31 \neq 25 \)[/tex]
- For [tex]\( (4, 24) \)[/tex]: [tex]\( 28 + 4 = 32 \neq 24 \)[/tex]
None of the pairs satisfy the equation [tex]\( 28 + x = y \)[/tex].
3. [tex]\( x - y = 28 \)[/tex]
Check if subtracting [tex]\(y\)[/tex] from [tex]\(x\)[/tex] gives 28 for each pair:
- For [tex]\( (1, 27) \)[/tex]: [tex]\( 1 - 27 = -26 \neq 28 \)[/tex]
- For [tex]\( (2, 26) \)[/tex]: [tex]\( 2 - 26 = -24 \neq 28 \)[/tex]
- For [tex]\( (3, 25) \)[/tex]: [tex]\( 3 - 25 = -22 \neq 28 \)[/tex]
- For [tex]\( (4, 24) \)[/tex]: [tex]\( 4 - 24 = -20 \neq 28 \)[/tex]
None of the pairs satisfy the equation [tex]\( x - y = 28 \)[/tex].
4. [tex]\( 28 - x = y \)[/tex]
Check if subtracting [tex]\(x\)[/tex] from 28 gives [tex]\(y\)[/tex] for each pair:
- For [tex]\( (1, 27) \)[/tex]: [tex]\( 28 - 1 = 27 \)[/tex]
- For [tex]\( (2, 26) \)[/tex]: [tex]\( 28 - 2 = 26 \)[/tex]
- For [tex]\( (3, 25) \)[/tex]: [tex]\( 28 - 3 = 25 \)[/tex]
- For [tex]\( (4, 24) \)[/tex]: [tex]\( 28 - 4 = 24 \)[/tex]
All pairs satisfy the equation [tex]\( 28 - x = y \)[/tex].
5. [tex]\( 28 - y = x \)[/tex]
Check if subtracting [tex]\(y\)[/tex] from 28 gives [tex]\(x\)[/tex] for each pair:
- For [tex]\( (1, 27) \)[/tex]: [tex]\( 28 - 27 = 1 \)[/tex]
- For [tex]\( (2, 26) \)[/tex]: [tex]\( 28 - 26 = 2 \)[/tex]
- For [tex]\( (3, 25) \)[/tex]: [tex]\( 28 - 25 = 3 \)[/tex]
- For [tex]\( (4, 24) \)[/tex]: [tex]\( 28 - 24 = 4 \)[/tex]
All pairs satisfy the equation [tex]\( 28 - y = x \)[/tex].
Therefore, the 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]
