To determine whether each equation describes the line passing through the points (4,6) and (6,2), we first need to find the slope of the line through these points.
The slope \( m \) is calculated as follows:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 6}{6 - 4} = \frac{-4}{2} = -2 \]
Now, we have the slope \( m = -2 \). We can use the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
Using the point (4,6):
\[ y - 6 = -2(x - 4) \]
Let's analyze each equation:
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Equation: \( y - 6 = -2(x - 4) \)
This is the correct equation we derived by using the point (4,6) with the slope of -2.
- Answer: Yes
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Equation: \( y - 4 = -2(x - 6) \)
If we simplify this equation using point (6,2):
\[ y - 2 = -2(x - 6), \] which gives us another form using the point (6,2) and also has the correct slope.
- Answer: Yes
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Equation: \( y - 2 = -2(x + 6) \)
This equation uses the point (6,2) but incorrectly refers to point (−6). Thus, it is not a correct representation of the line.
- Answer: No
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Equation: \( y - 2 = -2(x - 6) \)
This is another way to derive the equation using the point (6,2), having the slope -2.
- Answer: Yes
Now, we summarize the answers:
- \( y - 6 = -2(x - 4) \): Yes
- \( y - 4 = -2(x - 6) \): Yes
- \( y - 2 = -2(x + 6) \): No
- \( y - 2 = -2(x - 6) \): Yes