A line intersects the points $(8,2)$ and $(12,-10)$. What is the slope of this line in simplest form?
$m=[?]$
Answer (2)
Identify the coordinates of the two points: ( 8 , 2 ) and ( 12 , − 10 ) .
Apply the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the coordinates into the formula: m = 12 − 8 − 10 − 2 .
Simplify the expression to find the slope: m = − 3 . The final answer is − 3 .
Explanation
Understanding the Problem We are given two points on a line, ( 8 , 2 ) and ( 12 , − 10 ) , and we want to find the slope of this line in simplest form. The slope of a line passing through two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula: m = x 2 − x 1 y 2 − y 1 where m represents the slope of the line.
Applying the Slope Formula Let's identify the coordinates of the given points: ( x 1 , y 1 ) = ( 8 , 2 ) and ( x 2 , y 2 ) = ( 12 , − 10 ) .
Now, we substitute these values into the slope formula: m = 12 − 8 − 10 − 2
Simplifying the Expression Next, we simplify the expression: m = 4 − 12 m = − 3 So, the slope of the line is − 3 .
Final Answer The slope of the line in simplest form is − 3 .
Examples
Understanding the slope of a line is crucial in many real-world applications. For example, consider a ramp for wheelchair access. If the ramp rises 1 foot for every 3 feet of horizontal distance, the slope of the ramp is 3 1 . This slope helps ensure the ramp is not too steep, making it easier for wheelchair users to navigate. Similarly, in construction, the slope of a roof is essential for proper water runoff and structural integrity. A steeper slope allows water to drain more quickly, preventing leaks and damage to the building.
The slope of the line that intersects the points (8, 2) and (12, -10) is -3, indicating a downward slope as the x-axis increases.
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