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To determine which point-slope equations are correct for the line passing through the points [tex]\((5,6)\)[/tex] and [tex]\((1,1)\)[/tex], let's follow these steps:
1. Calculate the slope [tex]\( m \)[/tex] of the line passing through the points:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the given points [tex]\((x_1, y_1) = (5, 6)\)[/tex] and [tex]\((x_2, y_2) = (1, 1)\)[/tex], we find:
[tex]\[ m = \frac{1 - 6}{1 - 5} = \frac{-5}{-4} = \frac{5}{4} \][/tex]
2. Plug both points into each equation to verify which ones are correct:
Let's check each option one by one:
- Option A: [tex]\( y + 5 = \frac{4}{5} (x - 6) \)[/tex]
- For point [tex]\((1, 1)\)[/tex]:
[tex]\[ 1 + 5 = \frac{4}{5} (1 - 6) \implies 6 = \frac{4}{5} \times -5 \implies 6 = -4 \][/tex]
- For point [tex]\((5, 6)\)[/tex]:
[tex]\[ 6 + 5 = \frac{4}{5} (5 - 6) \implies 11 = \frac{4}{5} \times -1 \implies 11 = -0.8 \][/tex]
Since neither point satisfies the equation, Option A is incorrect.
- Option B: [tex]\( y - 1 = \frac{5}{4} (x + 1) \)[/tex]
- For point [tex]\((1, 1)\)[/tex]:
[tex]\[ 1 - 1 = \frac{5}{4} (1 + 1) \implies 0 = \frac{5}{4} \times 2 \implies 0 = 2.5 \][/tex]
- For point [tex]\((5, 6)\)[/tex]:
[tex]\[ 6 - 1 = \frac{5}{4} (5 + 1) \implies 5 = \frac{5}{4} \times 6 \implies 5 = 7.5 \][/tex]
Since neither point satisfies the equation, Option B is incorrect.
- Option C: [tex]\( y - 6 = \frac{5}{4} (x - 5) \)[/tex]
- For point [tex]\((1, 1)\)[/tex]:
[tex]\[ 1 - 6 = \frac{5}{4} (1 - 5) \implies -5 = \frac{5}{4} \times -4 \implies -5 = -5 \][/tex]
- For point [tex]\((5, 6)\)[/tex]:
[tex]\[ 6 - 6 = \frac{5}{4} (5 - 5) \implies 0 = \frac{5}{4} \times 0 \implies 0 = 0 \][/tex]
Since both points satisfy the equation, Option C is correct.
- Option D: [tex]\( y - 1 = \frac{5}{4} (x - 1) \)[/tex]
- For point [tex]\((1, 1)\)[/tex]:
[tex]\[ 1 - 1 = \frac{5}{4} (1 - 1) \implies 0 = \frac{5}{4} \times 0 \implies 0 = 0 \][/tex]
- For point [tex]\((5, 6)\)[/tex]:
[tex]\[ 6 - 1 = \frac{5}{4} (5 - 1) \implies 5 = \frac{5}{4} \times 4 \implies 5 = 5 \][/tex]
Since both points satisfy the equation, Option D is correct.
- Option E: [tex]\( y - 1 = \frac{4}{5} (x - 1) \)[/tex]
- For point [tex]\((1, 1)\)[/tex]:
[tex]\[ 1 - 1 = \frac{4}{5} (1 - 1) \implies 0 = \frac{4}{5} \times 0 \implies 0 = 0 \][/tex]
- For point [tex]\((5, 6)\)[/tex]:
[tex]\[ 6 - 1 = \frac{4}{5} (5 - 1) \implies 5 = \frac{4}{5} \times 4 \implies 5 = 3.2 \][/tex]
Since the second point does not satisfy the equation, Option E is incorrect.
- Option F: [tex]\( y + 6 = \frac{5}{4} (x + 5) \)[/tex]
- For point [tex]\((1, 1)\)[/tex]:
[tex]\[ 1 + 6 = \frac{5}{4} (1 + 5) \implies 7 = \frac{5}{4} \times 6 \implies 7 = 7.5 \][/tex]
- For point [tex]\((5, 6)\)[/tex]:
[tex]\[ 6 + 6 = \frac{5}{4} (5 + 5) \implies 12 = \frac{5}{4} \times 10 \implies 12 = 12.5 \][/tex]
Since neither point satisfies the equation, Option F is incorrect.
Thus, the correct point-slope equations are:
[tex]\[ \boxed{C \text{ and } D} \][/tex]
1. Calculate the slope [tex]\( m \)[/tex] of the line passing through the points:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the given points [tex]\((x_1, y_1) = (5, 6)\)[/tex] and [tex]\((x_2, y_2) = (1, 1)\)[/tex], we find:
[tex]\[ m = \frac{1 - 6}{1 - 5} = \frac{-5}{-4} = \frac{5}{4} \][/tex]
2. Plug both points into each equation to verify which ones are correct:
Let's check each option one by one:
- Option A: [tex]\( y + 5 = \frac{4}{5} (x - 6) \)[/tex]
- For point [tex]\((1, 1)\)[/tex]:
[tex]\[ 1 + 5 = \frac{4}{5} (1 - 6) \implies 6 = \frac{4}{5} \times -5 \implies 6 = -4 \][/tex]
- For point [tex]\((5, 6)\)[/tex]:
[tex]\[ 6 + 5 = \frac{4}{5} (5 - 6) \implies 11 = \frac{4}{5} \times -1 \implies 11 = -0.8 \][/tex]
Since neither point satisfies the equation, Option A is incorrect.
- Option B: [tex]\( y - 1 = \frac{5}{4} (x + 1) \)[/tex]
- For point [tex]\((1, 1)\)[/tex]:
[tex]\[ 1 - 1 = \frac{5}{4} (1 + 1) \implies 0 = \frac{5}{4} \times 2 \implies 0 = 2.5 \][/tex]
- For point [tex]\((5, 6)\)[/tex]:
[tex]\[ 6 - 1 = \frac{5}{4} (5 + 1) \implies 5 = \frac{5}{4} \times 6 \implies 5 = 7.5 \][/tex]
Since neither point satisfies the equation, Option B is incorrect.
- Option C: [tex]\( y - 6 = \frac{5}{4} (x - 5) \)[/tex]
- For point [tex]\((1, 1)\)[/tex]:
[tex]\[ 1 - 6 = \frac{5}{4} (1 - 5) \implies -5 = \frac{5}{4} \times -4 \implies -5 = -5 \][/tex]
- For point [tex]\((5, 6)\)[/tex]:
[tex]\[ 6 - 6 = \frac{5}{4} (5 - 5) \implies 0 = \frac{5}{4} \times 0 \implies 0 = 0 \][/tex]
Since both points satisfy the equation, Option C is correct.
- Option D: [tex]\( y - 1 = \frac{5}{4} (x - 1) \)[/tex]
- For point [tex]\((1, 1)\)[/tex]:
[tex]\[ 1 - 1 = \frac{5}{4} (1 - 1) \implies 0 = \frac{5}{4} \times 0 \implies 0 = 0 \][/tex]
- For point [tex]\((5, 6)\)[/tex]:
[tex]\[ 6 - 1 = \frac{5}{4} (5 - 1) \implies 5 = \frac{5}{4} \times 4 \implies 5 = 5 \][/tex]
Since both points satisfy the equation, Option D is correct.
- Option E: [tex]\( y - 1 = \frac{4}{5} (x - 1) \)[/tex]
- For point [tex]\((1, 1)\)[/tex]:
[tex]\[ 1 - 1 = \frac{4}{5} (1 - 1) \implies 0 = \frac{4}{5} \times 0 \implies 0 = 0 \][/tex]
- For point [tex]\((5, 6)\)[/tex]:
[tex]\[ 6 - 1 = \frac{4}{5} (5 - 1) \implies 5 = \frac{4}{5} \times 4 \implies 5 = 3.2 \][/tex]
Since the second point does not satisfy the equation, Option E is incorrect.
- Option F: [tex]\( y + 6 = \frac{5}{4} (x + 5) \)[/tex]
- For point [tex]\((1, 1)\)[/tex]:
[tex]\[ 1 + 6 = \frac{5}{4} (1 + 5) \implies 7 = \frac{5}{4} \times 6 \implies 7 = 7.5 \][/tex]
- For point [tex]\((5, 6)\)[/tex]:
[tex]\[ 6 + 6 = \frac{5}{4} (5 + 5) \implies 12 = \frac{5}{4} \times 10 \implies 12 = 12.5 \][/tex]
Since neither point satisfies the equation, Option F is incorrect.
Thus, the correct point-slope equations are:
[tex]\[ \boxed{C \text{ and } D} \][/tex]
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