Listen. Which pair of points has a slope of 4?
A. (0,-3),(1,1)
B. (0,4),(4,0)
C. (1,4),(-1,4)
D. (3,0),(-1,-1)
Answer (1)
The problem asks to find the pair of points that has a slope of 4. The slope between 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 .
Calculate the slope for the pair (0, -3) and (1, 1): m = 1 − 0 1 − ( − 3 ) = 4 .
Calculate the slope for the pair (0, 4) and (4, 0): m = 4 − 0 0 − 4 = − 1 .
Calculate the slope for the pair (1, 4) and (-1, 4): m = − 1 − 1 4 − 4 = 0 .
Calculate the slope for the pair (3, 0) and (-1, -1): m = − 1 − 3 − 1 − 0 = 4 1 .
The pair of points with a slope of 4 is ( 0 , − 3 ) , ( 1 , 1 ) .
Explanation
Understanding the Problem and Formula We are given four pairs of points and asked to find the pair with a slope of 4. The slope, m , between two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is calculated using the formula: m = x 2 − x 1 y 2 − y 1 We will calculate the slope for each pair of points and see which one equals 4.
Calculating Slope for (0, -3) and (1, 1) Let's calculate the slope for the first pair of points, (0, -3) and (1, 1): m = 1 − 0 1 − ( − 3 ) = 1 1 + 3 = 1 4 = 4 The slope for this pair is 4.
Calculating Slope for (0, 4) and (4, 0) Now, let's calculate the slope for the second pair of points, (0, 4) and (4, 0): m = 4 − 0 0 − 4 = 4 − 4 = − 1 The slope for this pair is -1.
Calculating Slope for (1, 4) and (-1, 4) Next, let's calculate the slope for the third pair of points, (1, 4) and (-1, 4): m = − 1 − 1 4 − 4 = − 2 0 = 0 The slope for this pair is 0.
Calculating Slope for (3, 0) and (-1, -1) Finally, let's calculate the slope for the fourth pair of points, (3, 0) and (-1, -1): m = − 1 − 3 − 1 − 0 = − 4 − 1 = 4 1 The slope for this pair is 4 1 .
Identifying the Correct Pair We are looking for a pair of points with a slope of 4. From our calculations, the pair (0, -3) and (1, 1) has a slope of 4.
Examples
Understanding slope is crucial in many real-world applications. For instance, civil engineers use slope to design roads and bridges, ensuring they are safe and efficient. A steeper slope might require more engine power for vehicles, while a gentler slope can improve fuel efficiency. In construction, the slope of a roof affects water runoff and the structural integrity of the building. By calculating and managing slopes, engineers can optimize designs for various environmental and functional factors.
