Find the slope of the line passing through the points $(6,3)$ and $(6,-3)$.
Answer (1)
Identify the coordinates of the two given points.
Apply the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the coordinates into the formula: m = 6 − 6 − 3 − 3 .
Simplify to find that the slope is undefined: undefined .
Explanation
Understanding the Problem We are given two points, ( 6 , 3 ) and ( 6 , − 3 ) , and we want to find the slope of the line that passes through them.
Recalling the Slope Formula 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.
Identifying Coordinates Let's identify the coordinates: x 1 = 6 , y 1 = 3 x 2 = 6 , y 2 = − 3
Calculating the Slope Now, substitute these values into the slope formula: m = 6 − 6 − 3 − 3 = 0 − 6 Since division by zero is undefined, the slope is undefined.
Interpreting the Result The slope is undefined because the line is vertical. A vertical line has the equation x = c , where c is a constant. In this case, the equation of the line is x = 6 .
Final Answer Therefore, the slope of the line passing through the points ( 6 , 3 ) and ( 6 , − 3 ) is undefined.
Examples
Imagine you're climbing a perfectly vertical ladder. The slope represents how steep the ladder is. In this case, a vertical ladder has an undefined slope because you aren't moving horizontally at all – just straight up! This concept applies in various fields, such as construction and engineering, where understanding slopes is crucial for building stable structures.
