Let's determine which of the given equations model the total yards of fabric (in yards, denoted by [tex]\(x\)[/tex] for red fabric and [tex]\(y\)[/tex] for blue fabric) that Sophie will buy. We will examine the given combinations and analyze each equation.
Here's the table:
[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 start by analyzing the equations one by one:
1. Equation [tex]\(x + y = 28\)[/tex]:
- For [tex]\((1, 27)\)[/tex]: [tex]\(1 + 27 = 28\)[/tex] (True)
- For [tex]\((2, 26)\)[/tex]: [tex]\(2 + 26 = 28\)[/tex] (True)
- For [tex]\((3, 25)\)[/tex]: [tex]\(3 + 25 = 28\)[/tex] (True)
- For [tex]\((4, 24)\)[/tex]: [tex]\(4 + 24 = 28\)[/tex] (True)
This equation holds true for all given pairs.
2. Equation [tex]\(28 + x = y\)[/tex]:
- For [tex]\((1, 27)\)[/tex]: [tex]\(28 + 1 = 29 \neq 27\)[/tex] (False)
- For [tex]\((2, 26)\)[/tex]: [tex]\(28 + 2 = 30 \neq 26\)[/tex] (False)
- For [tex]\((3, 25)\)[/tex]: [tex]\(28 + 3 = 31 \neq 25\)[/tex] (False)
- For [tex]\((4, 24)\)[/tex]: [tex]\(28 + 4 = 32 \neq 24\)[/tex] (False)
This equation does not hold true for any given pairs.
3. Equation [tex]\(x - y = 28\)[/tex]:
- For [tex]\((1, 27)\)[/tex]: [tex]\(1 - 27 = -26 \neq 28\)[/tex] (False)
- For [tex]\((2, 26)\)[/tex]: [tex]\(2 - 26 = -24 \neq 28\)[/tex] (False)
- For [tex]\((3, 25)\)[/tex]: [tex]\(3 - 25 = -22 \neq 28\)[/tex] (False)
- For [tex]\((4, 24)\)[/tex]: [tex]\(4 - 24 = -20 \neq 28\)[/tex] (False)
This equation does not hold true for any given pairs.
4. Equation [tex]\(28 - x = y\)[/tex]:
- For [tex]\((1, 27)\)[/tex]: [tex]\(28 - 1 = 27\)[/tex] (True)
- For [tex]\((2, 26)\)[/tex]: [tex]\(28 - 2 = 26\)[/tex] (True)
- For [tex]\((3, 25)\)[/tex]: [tex]\(28 - 3 = 25\)[/tex] (True)
- For [tex]\((4, 24)\)[/tex]: [tex]\(28 - 4 = 24\)[/tex] (True)
This equation holds true for all given pairs.
5. Equation [tex]\(28 - y = x\)[/tex]:
- For [tex]\((1, 27)\)[/tex]: [tex]\(28 - 27 = 1\)[/tex] (True)
- For [tex]\((2, 26)\)[/tex]: [tex]\(28 - 26 = 2\)[/tex] (True)
- For [tex]\((3, 25)\)[/tex]: [tex]\(28 - 25 = 3\)[/tex] (True)
- For [tex]\((4, 24)\)[/tex]: [tex]\(28 - 24 = 4\)[/tex] (True)
This equation holds true for all given pairs.
Based on the analysis:
[tex]\[
x + y = 28 \quad \text{(True)} \\
28 + x = y \quad \text{(False)} \\
x - y = 28 \quad \text{(False)} \\
28 - x = y \quad \text{(True)} \\
28 - y = x \quad \text{(True)}
\][/tex]
The equations that correctly model the total yards of fabric Sophie will buy are:
[tex]\[
x + y = 28 \\
28 - x = y \\
28 - y = x
\][/tex]
Thus, the equations that apply are:
[tex]\((1) \, x + y = 28\)[/tex], [tex]\((4) \, 28 - x = y\)[/tex], and [tex]\((5) \, 28 - y = x\)[/tex].
