A software designer is mapping the streets for a new racing game. All of the streets are dep equation of the lane passing through A and B is $-7 x+3 y=-21.5$. What is the equation of the
A. $-3 x+4 y=3$
B. $3 x+7 y=63$
C. $2 x+y=20$
D. $7 x+3 y=70$
Answer (1)
The problem provides an equation − 7 x + 3 y = − 21.5 and asks to find an equivalent equation from the given options. After analyzing the options and testing points, the closest option is 7 x + 3 y = 70 , although it's not exactly equivalent. There might be an error in the question or the options. The final answer is D.
Explanation
Understanding the Problem We are given the equation of a lane as − 7 x + 3 y = − 21.5 . We need to find an equivalent equation from the given options.
Eliminating the Decimal First, let's multiply the given equation by 2 to eliminate the decimal: − 7 x + 3 y = − 21.5
2 ( − 7 x + 3 y ) = 2 ( − 21.5 )
− 14 x + 6 y = − 43
Analyzing the Options Now, let's examine the given options to see if any of them are equivalent to the given equation or a multiple of it.
A. − 3 x + 4 y = 3 B. 3 x + 7 y = 63 C. 2 x + y = 20 D. 7 x + 3 y = 70
Testing a Point We can multiply option D by -1 to get − 7 x − 3 y = − 70 . This is not a multiple of the original equation. However, we can test a point to see if it satisfies both equations. Let's find a point on the original line. If we set y = 0 , we get − 7 x = − 21.5 , so x = 7 21.5 = 14 43 . The point ( 14 43 , 0 ) should satisfy the correct option.
Checking Option D Let's test option D: 7 x + 3 y = 70 . Plugging in the point ( 14 43 , 0 ) , we get: 7 ( 14 43 ) + 3 ( 0 ) = 70
2 43 = 70
21.5 = 70 This is not true, so option D is incorrect.
Testing Options with a Point Let's try to find a point that satisfies − 7 x + 3 y = − 21.5 . If we let x = 10 , then − 7 ( 10 ) + 3 y = − 21.5 , so − 70 + 3 y = − 21.5 , which means 3 y = 48.5 , and y = 3 48.5 = 6 97 . So the point ( 10 , 6 97 ) is on the line. Now we test the options.
A. − 3 x + 4 y = 3 . Plugging in the point, we get − 3 ( 10 ) + 4 ( 6 97 ) = − 30 + 3 194 = 3 − 90 + 194 = 3 104 = 3 .
B. 3 x + 7 y = 63 . Plugging in the point, we get 3 ( 10 ) + 7 ( 6 97 ) = 30 + 6 679 = 6 180 + 679 = 6 859 = 63 .
C. 2 x + y = 20 . Plugging in the point, we get 2 ( 10 ) + 6 97 = 20 + 6 97 = 6 120 + 97 = 6 217 = 20 .
D. 7 x + 3 y = 70 . Plugging in the point, we get 7 ( 10 ) + 3 ( 6 97 ) = 70 + 2 97 = 2 140 + 97 = 2 237 = 70 .
Trying Another Point Let's try to manipulate the original equation to match one of the options. Notice that option D has 7 x + 3 y . Our equation has − 7 x + 3 y . If we choose x = 7 , then − 7 ( 7 ) + 3 y = − 21.5 , so − 49 + 3 y = − 21.5 , which gives 3 y = 27.5 , so y = 3 27.5 = 6 55 . So the point ( 7 , 6 55 ) is on the line. Let's test option D with this point: 7 ( 7 ) + 3 ( 6 55 ) = 49 + 2 55 = 2 98 + 55 = 2 153 = 76.5 = 70 .
Final Analysis and Conclusion Let's analyze option D further. If we have 7 x + 3 y = 70 , we can rewrite the original equation as − 7 x + 3 y = − 21.5 . Adding these two equations, we get 6 y = 48.5 , so y = 6 48.5 = 12 97 . Substituting this into the first equation, 7 x + 3 ( 12 97 ) = 70 , so 7 x + 4 97 = 70 , which means 7 x = 70 − 4 97 = 4 280 − 97 = 4 183 , so x = 28 183 . Thus, the intersection of the two lines is ( 28 183 , 12 97 ) . Since the question states that the given equation is the equation of the lane passing through A and B, and asks for the equation of the lane, it seems there might be an error in the question or the options. However, based on the given options, the closest one is D.
Examples
In designing a racing game, mapping streets involves defining their paths mathematically. The equation of a lane represents its trajectory, and understanding these equations helps in creating realistic and challenging race tracks. For instance, knowing the equations of lanes allows developers to simulate vehicle movement, calculate distances, and design turns, making the game more engaging and accurate.
