Find the slope of the line that passes through these two points: $(4,1)$ and $(8,4)$.
Is the slope positive or negative?
A. positive
B. negative
Slope Formula: $\frac{y_2-y_1}{x_2-x_1}$
Answer (1)
Assign coordinates: ( x 1 , y 1 ) = ( 4 , 1 ) and ( x 2 , y 2 ) = ( 8 , 4 ) .
Apply the slope formula: m = x 2 − x 1 y 2 − y 1 .
Calculate the slope: m = 8 − 4 4 − 1 = 4 3 .
Determine the sign: The slope is positive. 4 3
Explanation
Understanding the Problem We are given two points, ( 4 , 1 ) and ( 8 , 4 ) , and we want to find the slope of the line that passes through them. We are also given the slope formula: m = x 2 − x 1 y 2 − y 1 .
Assigning Coordinates Let's assign the coordinates: ( x 1 , y 1 ) = ( 4 , 1 ) and ( x 2 , y 2 ) = ( 8 , 4 ) . Now, we can plug these values into the slope formula.
Calculating the Slope Using the slope formula, we have: m = 8 − 4 4 − 1 = 4 3 = 0.75 So, the slope of the line is 4 3 or 0.75 .
Determining the Sign of the Slope Since the slope 4 3 is a positive number, the slope is positive.
Final Answer The slope of the line that passes through the points ( 4 , 1 ) and ( 8 , 4 ) is 4 3 , and the slope is positive. Therefore, the answer to the first question is 4 3 , and the answer to the second question is (a) positive.
Examples
Understanding the slope is crucial in many real-world applications. For instance, when designing a ramp for accessibility, the slope determines how gradually someone can ascend. A smaller slope means a more gradual and easier climb. Similarly, in construction, the slope of a roof affects water runoff and structural integrity. Calculating the slope between two points helps engineers and architects ensure safety and functionality in their designs. This concept is also used in economics to represent rates of change, such as the marginal cost or revenue.
