A line has a slope of $-\frac{3}{5}$. Which ordered pairs could be points on a parallel line? Select two options.
(-8,8) and (2,2)
(-5,-1) and (0,2)
(-3,6) and (6,-9)
(-2,1) and (3,-2)
(0,2) and (5,5)
Answer (2)
Calculate the slope between each pair of points using the formula m = x 2 − x 1 y 2 − y 1 .
Compare the calculated slopes with the given slope of − 5 3 = − 0.6 .
Identify the pairs with a slope of − 5 3 .
The pairs of ordered pairs that lie on lines parallel to the given line are ( − 8 , 8 ) and ( 2 , 2 ) , and ( − 2 , 1 ) and ( 3 , − 2 ) .
Explanation
Understanding the Problem We are given a line with a slope of − 5 3 , which is equal to -0.6. We need to find two pairs of ordered pairs that lie on lines parallel to the given line. Parallel lines have the same slope. Therefore, we need to calculate the slopes between the given pairs of points and identify the pairs with a slope of -0.6.
Using the Slope Formula We will use the slope formula to calculate the slope between each pair of points. The slope formula is given by: m = x 2 − x 1 y 2 − y 1 where ( x 1 , y 1 ) and ( x 2 , y 2 ) are the coordinates of the two points.
Calculating Slopes
Pair 1: ( − 8 , 8 ) and ( 2 , 2 ) m = 2 − ( − 8 ) 2 − 8 = 10 − 6 = − 0.6 = − 5 3
Pair 2: ( − 5 , − 1 ) and ( 0 , 2 ) m = 0 − ( − 5 ) 2 − ( − 1 ) = 5 3 = 0.6
Pair 3: ( − 3 , 6 ) and ( 6 , − 9 ) m = 6 − ( − 3 ) − 9 − 6 = 9 − 15 = − 3 5 ≈ − 1.67
Pair 4: ( − 2 , 1 ) and ( 3 , − 2 ) m = 3 − ( − 2 ) − 2 − 1 = 5 − 3 = − 0.6
Pair 5: ( 0 , 2 ) and ( 5 , 5 ) m = 5 − 0 5 − 2 = 5 3 = 0.6
Analyzing the Results From the calculations above, we can see that:
Pair 1 has a slope of − 5 3 = − 0.6 .
Pair 2 has a slope of 5 3 = 0.6 .
Pair 3 has a slope of − 3 5 ≈ − 1.67 .
Pair 4 has a slope of − 5 3 = − 0.6 .
Pair 5 has a slope of 5 3 = 0.6 .
Final Answer Since parallel lines have the same slope, we are looking for pairs with a slope of − 5 3 = − 0.6 . Therefore, the two pairs of ordered pairs that could be points on a parallel line are:
( − 8 , 8 ) and ( 2 , 2 )
( − 2 , 1 ) and ( 3 , − 2 )
Examples
Understanding parallel lines is crucial in architecture and design. For example, when designing a building, architects use parallel lines to create a sense of order and stability. The concept of slope is also essential in determining the steepness of roofs or ramps, ensuring they meet safety standards and aesthetic requirements. By applying these mathematical principles, architects can create visually appealing and structurally sound designs.
The two pairs of ordered pairs that could represent points on a parallel line to the given slope of − 5 3 are ( − 8 , 8 ) and ( 2 , 2 ) and ( − 2 , 1 ) and ( 3 , − 2 ) .
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