To find the linear equation in slope-intercept form (y = mx + b) using the point-slope formula, we first need to determine the slope (m) between the two given points, (8, 5) and (-4, -1).
Step 1: Calculate the slope (m)
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \( (x_1, y_1) = (8, 5) \) and \( (x_2, y_2) = (-4, -1) \):
\[ m = \frac{-1 - 5}{-4 - 8} = \frac{-6}{-12} = \frac{1}{2} \]
Step 2: Use point-slope form
The point-slope form of the equation of a line is given by:
\[ y - y_1 = m(x - x_1) \]
We can use either point; let's use the point (8, 5):
\[ y - 5 = \frac{1}{2}(x - 8) \]
Step 3: Simplify to slope-intercept form
Now, we simplify this equation to get it in slope-intercept form (y = mx + b):
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Distribute \( \frac{1}{2} \): \[ y - 5 = \frac{1}{2}x - 4 \]
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Add 5 to both sides: \[ y = \frac{1}{2}x - 4 + 5 \]
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Combine the constants: \[ y = \frac{1}{2}x + 1 \]
Conclusion
The linear equation in slope-intercept form is:
\[ y = \frac{1}{2}x + 1 \]