Find the slope of the line passing through the points $(-4,-7)$ and $(-4,5)$.
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 = − 4 − ( − 4 ) 5 − ( − 7 ) .
Simplify the expression: m = 0 12 , which is undefined. The final answer is undefined .
Explanation
Understanding the Problem We are given two points, ( − 4 , − 7 ) and ( − 4 , 5 ) , 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 = − 4 , y 1 = − 7 x 2 = − 4 , y 2 = 5
Substituting Values Now, substitute these values into the slope formula: m = − 4 − ( − 4 ) 5 − ( − 7 )
Simplifying the Expression Simplify the expression: m = − 4 + 4 5 + 7 = 0 12
Determining the Slope Since division by zero is undefined, the slope of the line is undefined. This means the line is vertical.
Examples
Imagine you're climbing a perfectly vertical ladder. You move upwards (change in y), but you don't move horizontally at all (change in x is zero). Trying to calculate the 'steepness' (slope) of this ladder results in division by zero, which is undefined. This is analogous to finding the slope of a vertical line. Understanding undefined slopes helps in various applications, such as designing structures, understanding rates of change, and analyzing graphs.
