To find the equations of a line that passes through the points (3, -4) and (−11, 3), we first need to determine the slope (m) of the line using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let (x₁, y₁) = (3, -4) and (x₂, y₂) = (−11, 3):
\[ m = \frac{3 - (-4)}{-11 - 3} = \frac{3 + 4}{-11 - 3} = \frac{7}{-14} = -\frac{1}{2} \]
So, the slope of the line is \(-\frac{1}{2}\).
Next, we can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
Using the point (3, -4) and the slope \(-\frac{1}{2}\):
\[ y - (-4) = -\frac{1}{2}(x - 3) \]
This simplifies to:
\[ y + 4 = -\frac{1}{2}(x - 3) \]
Now let's check the given options.
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Option 1: \(y + 4 = 2(x - 3)\)
- This cannot be correct as the slope is positive (2) instead of negative.
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Option 2: \(y - 3 = -\frac{2}{1}(x + 11)\)
- This simplifies to \(y - 3 = -2(x + 11)\).
- Simplifying gives \(y - 3 = -2x - 22\), thus, \(y = -2x - 19\) which is not correct because it possesses a different slope (\(-2\)).
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Option 3: \(y + 4 = -\frac{2}{1}(x - 3)\)
- This simplifies to \(y + 4 = -2(x - 3)\).
- Rearranging gives: \(y + 4 = -2x + 6\), or \(y = -2x + 2\) which also does not match our calculated slope.
Since both Options 2 and 3 use slopes that are incorrect compared to our original calculation, the correct expression derived from our calculations is:
\[ y + 4 = -\frac{1}{2}(x - 3) \]
Hence the correct answers should represent the same line and would only be the rearrangement of the derived expression, and any distinct derivation yielding \(-\frac{1}{2}\) in its simplest form with the same y-intercept should be validated against the outputs.
In conclusion, since none of the given options is correct, it seems there is an inconsistency in the options provided. The correct equation should have a slope of \(-\frac{1}{2}\) and pass through the given points.