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Sagot :
To determine which equations accurately represent the data in the given table, we need to substitute each data pair [tex]\((x, y)\)[/tex] into each equation and check if the equation holds true.
Let's go through each equation step-by-step:
1. First Equation: [tex]\( y - 6 = \frac{-5}{4}(x + 2) \)[/tex]
We will substitute each [tex]\((x, y)\)[/tex] pair:
- For [tex]\((x = -2, y = 6)\)[/tex]:
[tex]\[ 6 - 6 = \frac{-5}{4}(-2 + 2) \Rightarrow 0 = 0 \quad \text{(This is true)} \][/tex]
Since this pair satisfies the equation, it represents the data.
2. Second Equation: [tex]\( y - 2 = \frac{-5}{4}(x - 1) \)[/tex]
Let's substitute each [tex]\((x, y)\)[/tex] pair:
- For [tex]\((x = 0, y = 3.5)\)[/tex]:
[tex]\[ 3.5 - 2 = \frac{-5}{4}(0 - 1) \Rightarrow 1.5 = 1.25 \quad \text{(This is false)} \][/tex]
- For [tex]\((x = 2, y = 1)\)[/tex]:
[tex]\[ 1 - 2 = \frac{-5}{4}(2 - 1) \Rightarrow -1 = -1.25 \quad \text{(This is false)} \][/tex]
None of the pairs satisfy the equation, so it does not represent the data.
3. Third Equation: [tex]\( y + 2 = \frac{-5}{4}(x - 6) \)[/tex]
Substituting each [tex]\((x, y)\)[/tex] pair:
- For [tex]\((x = 0, y = 3.5)\)[/tex]:
[tex]\[ 3.5 + 2 = \frac{-5}{4}(0 - 6) \Rightarrow 5.5 = 7.5 \quad \text{(This is false)} \][/tex]
- For [tex]\((x = 2, y = 1)\)[/tex]:
[tex]\[ 1 + 2 = \frac{-5}{4}(2 - 6) \Rightarrow 3 = 5 \quad \text{(This is false)} \][/tex]
None of the pairs satisfy the equation, so it does not represent the data.
4. Fourth Equation: [tex]\( y - 1 = \frac{-5}{4}(x - 2) \)[/tex]
Substituting each [tex]\((x, y)\)[/tex] pair:
- For [tex]\((x = 2, y = 1)\)[/tex]:
[tex]\[ 1 - 1 = \frac{-5}{4}(2 - 2) \Rightarrow 0 = 0 \quad \text{(This is true)} \][/tex]
Since this pair satisfies the equation, it represents the data.
5. Fifth Equation: [tex]\( y - 3.5 = -1.25 x \)[/tex]
Substituting each [tex]\((x, y)\)[/tex] pair:
- For [tex]\((x = 0, y = 3.5)\)[/tex]:
[tex]\[ 3.5 - 3.5 = -1.25(0) \Rightarrow 0 = 0 \quad \text{(This is true)} \][/tex]
Since this pair satisfies the equation, it represents the data.
In summary, the equations that represent the data in the table are:
[tex]\[ y - 6 = \frac{-5}{4}(x + 2) \][/tex]
[tex]\[ y - 1 = \frac{-5}{4}(x - 2) \][/tex]
[tex]\[ y - 3.5 = -1.25 x \][/tex]
Thus, the equations that apply are the first, fourth, and fifth equations.
Let's go through each equation step-by-step:
1. First Equation: [tex]\( y - 6 = \frac{-5}{4}(x + 2) \)[/tex]
We will substitute each [tex]\((x, y)\)[/tex] pair:
- For [tex]\((x = -2, y = 6)\)[/tex]:
[tex]\[ 6 - 6 = \frac{-5}{4}(-2 + 2) \Rightarrow 0 = 0 \quad \text{(This is true)} \][/tex]
Since this pair satisfies the equation, it represents the data.
2. Second Equation: [tex]\( y - 2 = \frac{-5}{4}(x - 1) \)[/tex]
Let's substitute each [tex]\((x, y)\)[/tex] pair:
- For [tex]\((x = 0, y = 3.5)\)[/tex]:
[tex]\[ 3.5 - 2 = \frac{-5}{4}(0 - 1) \Rightarrow 1.5 = 1.25 \quad \text{(This is false)} \][/tex]
- For [tex]\((x = 2, y = 1)\)[/tex]:
[tex]\[ 1 - 2 = \frac{-5}{4}(2 - 1) \Rightarrow -1 = -1.25 \quad \text{(This is false)} \][/tex]
None of the pairs satisfy the equation, so it does not represent the data.
3. Third Equation: [tex]\( y + 2 = \frac{-5}{4}(x - 6) \)[/tex]
Substituting each [tex]\((x, y)\)[/tex] pair:
- For [tex]\((x = 0, y = 3.5)\)[/tex]:
[tex]\[ 3.5 + 2 = \frac{-5}{4}(0 - 6) \Rightarrow 5.5 = 7.5 \quad \text{(This is false)} \][/tex]
- For [tex]\((x = 2, y = 1)\)[/tex]:
[tex]\[ 1 + 2 = \frac{-5}{4}(2 - 6) \Rightarrow 3 = 5 \quad \text{(This is false)} \][/tex]
None of the pairs satisfy the equation, so it does not represent the data.
4. Fourth Equation: [tex]\( y - 1 = \frac{-5}{4}(x - 2) \)[/tex]
Substituting each [tex]\((x, y)\)[/tex] pair:
- For [tex]\((x = 2, y = 1)\)[/tex]:
[tex]\[ 1 - 1 = \frac{-5}{4}(2 - 2) \Rightarrow 0 = 0 \quad \text{(This is true)} \][/tex]
Since this pair satisfies the equation, it represents the data.
5. Fifth Equation: [tex]\( y - 3.5 = -1.25 x \)[/tex]
Substituting each [tex]\((x, y)\)[/tex] pair:
- For [tex]\((x = 0, y = 3.5)\)[/tex]:
[tex]\[ 3.5 - 3.5 = -1.25(0) \Rightarrow 0 = 0 \quad \text{(This is true)} \][/tex]
Since this pair satisfies the equation, it represents the data.
In summary, the equations that represent the data in the table are:
[tex]\[ y - 6 = \frac{-5}{4}(x + 2) \][/tex]
[tex]\[ y - 1 = \frac{-5}{4}(x - 2) \][/tex]
[tex]\[ y - 3.5 = -1.25 x \][/tex]
Thus, the equations that apply are the first, fourth, and fifth equations.
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