To derive the equation of the line in slope-intercept form (y = mx + b) based on the points (1, 3) and (0, -2), we will follow these steps:
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Determine the slope (m) of the line. The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let's use the points (1, 3) and (0, -2):
\((x_1, y_1) = (1, 3)\) and \((x_2, y_2) = (0, -2)\)
Substitute the values into the slope formula:
\[ m = \frac{-2 - 3}{0 - 1} = \frac{-5}{-1} = 5 \]
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Substitute the slope (m) and one of the points into the point-slope form of the equation. We can use point (0, -2) to find \(b\):
The slope-intercept form of an equation is:
\[ y = mx + b \]
Plug in the slope \(m = 5\) and the point (0, -2):
\(-2 = 5(0) + b\)
Simplifying this, we get:
\(-2 = 0 + b\)
Thus:
\[ b = -2 \]
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Write the final equation in slope-intercept form:
\[ y = 5x - 2 \]
Thus, the equation of the line in slope-intercept form is:
\[ y = 5x - 2 \]