Find the slope of the line passing through the points $(4,3)$ and $(4,-7)$.
Answer (1)
The slope formula is m = x 2 − x 1 y 2 − y 1 .
Substitute the coordinates of the given points into the formula: m = 4 − 4 − 7 − 3 .
Simplify the expression: m = 0 − 10 .
Since division by zero is undefined, the slope is undefined: undefined .
Explanation
Understanding the Problem We are given two points, ( 4 , 3 ) and ( 4 , − 7 ) , 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
Identifying Coordinates Let's identify the coordinates: x 1 = 4 , y 1 = 3 x 2 = 4 , y 2 = − 7
Applying the Slope Formula Now, we substitute these values into the slope formula: m = 4 − 4 − 7 − 3 = 0 − 10
Determining the Slope Since division by zero is undefined, the slope of the line is undefined. This means the line is vertical.
Final Answer The slope of the line passing through the points ( 4 , 3 ) and ( 4 , − 7 ) is undefined.
Examples
Imagine you're designing a very steep ramp. If the ramp is perfectly vertical, it's impossible to walk or roll anything up it. In mathematical terms, the slope is undefined because the horizontal change is zero. Understanding undefined slopes helps in real-world scenarios like designing structures or understanding rates of change that become infinitely large.
