To determine the correct system of equations based on the given points (2, 7) and (-1, 1), we need to find the equations of lines that pass through these points.
Let's first find the equation of the line that goes through the two points.
- Finding the slope (m):
Using the formula for the slope: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1) = (2, 7)\) and \((x_2, y_2) = (-1, 1)\): \[ m = \frac{1 - 7}{-1 - 2} = \frac{-6}{-3} = 2 \]
- Using point-slope form:
Using the point-slope form of the equation \(y - y_1 = m(x - x_1)\) with the point (2, 7): \[ y - 7 = 2(x - 2) \] Expanding this gives: \[ y - 7 = 2x - 4 \quad \Rightarrow \quad y = 2x + 3 \]
Now, we can also find the second line that should intersect the same points.
- Finding the second line
Using the point (−1, 1): Referencing \(y - 1 = m(x + 1)\) where we need to find the angle, we can assume that it will have a different slope. If we use the information we found earlier about the first line, we already know the slope is 2.
So we have: The first line is \(y = 2x + 3\) and now we also could guess a second line to intersect, starting from the y-intercept as such that it will create a system of two lines.
Upon analyzing the given options:
- Option A has \(y = 2x + 7\) which doesn't match our derived line \(y = 2x + 3\).
- Option B has \(y = 2x - 3\) which matches with \(2x\) but does not meet through the points.
- Option C contains \(y = 3x + 4\) does not fulfill our points either.
- Option D is incorrect too.
After checking, none of the answer choices seem to contain both correct lines simultaneously.
Thus, based on what we evaluated at the points:
- The first equation we derived is \(y = 2x + 3\).
Thus it seems the given choices might have been misreferenced as typically it would be values intersecting and about to represented.
Nevertheless, for lines intersecting through the points with respect; double-check with the graphical representation of what was calculated -- for future references.
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