To find the linear equation of a line that passes through the points (−1, 7) and (2, 4) using the point-slope form method, we first need to determine the slope of the line, and then apply the point-slope formula \( y - y_1 = m(x - x_1) \).
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Find the slope (m): The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \((−1, 7)\) and \((2, 4)\):
\[ m = \frac{4 - 7}{2 - (-1)} = \frac{-3}{3} = -1 \]
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Using the point-slope form with one of the points: The point-slope form of a line is:
\[ y - y_1 = m(x - x_1) \]
Using the point (2, 4):
\[ y - 4 = -1(x - 2) \]
(This matches with one of the provided options.)
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Alternatively, we could also use the point (−1, 7):
\[ y - 7 = -1(x - (-1)) = -1(x + 1) \]
(This also matches with another provided option.)
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Verifying the provided statements:
- \(y = x + 6\): Incorrect.
- \(7 = –1(–1) + b\): This appears incorrect as there is no context, but it seems to be referring to a b-intercept calculation.
- \(y - 4 = -1 (x - 2)\): Correct, corresponds to \( m = -1 \) using point (2,4).
- \(y - 7 = -1 (x - (-1))\): Correct, corresponds to \( m = -1 \) using point (-1,7).
- \(y - 2 = x - 4\): Incorrect.
- \(y = -x + 6\): Correct, after simplifying either of our equations will yield \(y = -x + 6\).
Therefore, the correct steps are:
- \( y - 4 = -1 (x - 2) \)
- \( y - 7 = -1 (x - (-1)) \)
- \( y = -x + 6 \) (upon deriving from the point-slope form and simplifying)
In summary, the correct options are the last three statements.