To solve this question, we need to analyze the given data and observe the relationship between the quantities of red and blue fabric that Sophie might buy. The table provides the following combinations of red and blue fabric:
| Red fabric (yards), [tex]\( x \)[/tex] | Blue fabric (yards), [tex]\( y \)[/tex] |
|-----------------------------|------------------------------|
| 1 | 27 |
| 2 | 26 |
| 3 | 25 |
| 4 | 24 |
By examining the table, we can see that as [tex]\( x \)[/tex] increases by 1, [tex]\( y \)[/tex] decreases by 1. This suggests a fixed total amount of fabric that Sophie might buy each time, which is the sum of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. We can use this observation to derive the following equations:
1. [tex]\( x + y = 28 \)[/tex]
- This equation directly shows that the sum of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is always 28.
2. [tex]\( 28 - x = y \)[/tex]
- This equation can be derived from [tex]\( x + y = 28 \)[/tex]. Rearranging the equation, we get [tex]\( y = 28 - x \)[/tex].
3. [tex]\( x = 28 - y \)[/tex]
- Similarly, this equation can be derived from [tex]\( x + y = 28 \)[/tex]. Rearranging the equation, we get [tex]\( x = 28 - y \)[/tex].
4. [tex]\( 28 = x + y \)[/tex]
- This equation is a rearrangement of [tex]\( x + y = 28 \)[/tex] and is equivalent to the first equation.
Thus, the equations that model the total yards of fabric Sophie will buy are:
- [tex]\( x + y = 28 \)[/tex]
- [tex]\( 28 - x = y \)[/tex]
- [tex]\( x = 28 - y \)[/tex]
- [tex]\( 28 = x + y \)[/tex]
The other given equations do not correctly represent the relationship shown in the table:
- [tex]\( 28 + x = y \)[/tex] would suggest an additional amount added to [tex]\( x \)[/tex], which doesn't match the observed total sum.
- [tex]\( x - y = 28 \)[/tex] suggests [tex]\( x \)[/tex] is always 28 more than [tex]\( y \)[/tex], which contradicts the data.
- [tex]\( 28 \cdot y = x \)[/tex] implies a multiplicative relationship that isn’t observed in the table.
So, the correct equations are:
- [tex]\( x + y = 28 \)[/tex]
- [tex]\( 28 - x = y \)[/tex]
- [tex]\( x = 28 - y \)[/tex]
- [tex]\( 28 = x + y \)[/tex]
