Find the slope of the given line, if it is defined.
The line through $(0,-1)$ and $(-6,-10)$.
Answer (1)
Identify the coordinates of the two points: ( 0 , − 1 ) and ( − 6 , − 10 ) .
Apply the slope formula: m = x 2 − x 1 y 2 − y 1 .
Substitute the coordinates into the formula: m = − 6 − 0 − 10 − ( − 1 ) .
Simplify to find the slope: m = 1.5 . The slope of the line is 1.5 .
Explanation
Understanding the Problem We are given two points on a line, ( 0 , − 1 ) and ( − 6 , − 10 ) , and we want to find the slope of this line. 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 = 0 , y 1 = − 1 x 2 = − 6 , y 2 = − 10
Applying the Slope Formula Now, we substitute these values into the slope formula: m = − 6 − 0 − 10 − ( − 1 )
Simplifying the Expression Simplify the expression: m = − 6 − 10 + 1 = − 6 − 9 = 2 3
Finding the Slope Convert the fraction to a decimal: m = 2 3 = 1.5 Therefore, the slope of the line is 1.5.
Examples
Understanding the slope is crucial in many real-world applications. For instance, in construction, the slope of a ramp determines its steepness and accessibility. In economics, the slope of a supply or demand curve indicates how sensitive the quantity supplied or demanded is to changes in price. In physics, the slope of a velocity-time graph represents acceleration. Knowing how to calculate and interpret slope allows us to analyze and design systems effectively in various fields.
