Find the slope of the line containing the points $(4,-7)$ and $(6,-8)$.
A. 8
B. $-3 / 2$
C. $-1 / 2$
D. -2
Answer (2)
Use the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the given points ( 4 , − 7 ) and ( 6 , − 8 ) into the formula: m = 6 − 4 − 8 − ( − 7 ) .
Simplify the expression: m = 2 − 1 .
The slope of the line is − 2 1 .
Explanation
Understanding the Problem We are given two points, ( 4 , − 7 ) and ( 6 , − 8 ) , and we want to find the slope of the line that passes through them. 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
Applying the Slope Formula In our case, we have ( x 1 , y 1 ) = ( 4 , − 7 ) and ( x 2 , y 2 ) = ( 6 , − 8 ) . Plugging these values into the slope formula, we get: m = 6 − 4 − 8 − ( − 7 )
Calculating the Slope Now, we simplify the expression: m = 6 − 4 − 8 + 7 = 2 − 1 So, the slope of the line is − 2 1 .
Final Answer Therefore, the slope of the line containing the points ( 4 , − 7 ) and ( 6 , − 8 ) is − 2 1 .
Examples
Imagine you're hiking on a trail, and you want to describe how steep the path is. If you know two points on the trail, you can calculate the slope between those points. For example, if at 4 meters horizontally, you are -7 meters in altitude, and at 6 meters horizontally, you are -8 meters in altitude, the slope is -1/2. This means that for every 2 meters you move horizontally, you descend 1 meter vertically. Understanding slope helps in many real-world scenarios, such as designing roads, ramps, or even understanding the steepness of a roof.
The slope of the line containing the points (4, -7) and (6, -8) is -1/2. This is calculated using the slope formula, which gives the change in y over the change in x. Therefore, the correct answer is option C: -1/2.
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