What is the slope of the line that passes through the points $(4,0)$ and $(-1,1)$?
Answer (1)
Identify the two points: ( 4 , 0 ) and ( − 1 , 1 ) .
Apply the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the coordinates: m = − 1 − 4 1 − 0 .
Calculate the slope: m = − 5 1 .
The slope of the line is − 5 1 .
Explanation
Understanding the Problem We are given two points, ( 4 , 0 ) and ( − 1 , 1 ) , and we want to find the slope of the line that passes through these points.
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
Calculating the Slope Let's identify our points: ( x 1 , y 1 ) = ( 4 , 0 ) and ( x 2 , y 2 ) = ( − 1 , 1 ) . Now, we substitute these values into the slope formula: m = − 1 − 4 1 − 0 = − 5 1 = − 5 1
Final Answer Therefore, the slope of the line that passes through the points ( 4 , 0 ) and ( − 1 , 1 ) is − 5 1 .
Examples
Understanding the slope of a line is crucial in many real-world applications. For example, consider a ramp designed for wheelchair access. The slope of the ramp determines how easy or difficult it is to use. A steeper slope (larger positive number) requires more effort to ascend, while a gentler slope (smaller positive number or even a negative slope if descending) is easier. Civil engineers use slope calculations to ensure ramps and roads are safe and accessible.
