To find the equation of the line segment that connects the points given, we can analyze the coordinates of the points. It seems that the points you provided form a linear progression. We'll consider the beginning point (0, 6) and the end point (8, 10), as well as the midpoint points (2, 7), (4, 8), and (6, 9).
The equation of a line can be determined using the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
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Calculate the slope \( m \): The slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] We can use the first point \((0, 6)\) and the last point \((8, 10)\): \[ m = \frac{10 - 6}{8 - 0} = \frac{4}{8} = \frac{1}{2} \]
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Find the y-intercept \( b \): Using the slope-intercept form \( y = mx + b \) and substituting one of the points, such as \((0, 6)\): \[ 6 = \frac{1}{2}(0) + b \] Thus, \( b = 6 \).
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Write the equation: Now that we have \( m \) and \( b \), we can write the equation of the line: \[ y = \frac{1}{2}x + 6 \]
Now we can also check the intermediary points:
- For \( x = 2 \): \[ y = \frac{1}{2}(2) + 6 = 1 + 6 = 7 \]
- For \( x = 4 \): \[ y = \frac{1}{2}(4) + 6 = 2 + 6 = 8 \]
- For \( x = 6 \): \[ y = \frac{1}{2}(6) + 6 = 3 + 6 = 9 \]
All of these calculations confirm that the points \((2, 7)\), \((4, 8)\), and \((6, 9)\) lie on the line defined by \( y = \frac{1}{2}x + 6 \).
Therefore, the equation of the line connecting these points is: \[ y = \frac{1}{2}x + 6 \]