$3x + 2y + 8 = 0$
The taxi fare in a city is charged as per the rates stated below:
Rate for the first kilometer of journey is Rs 5 and the rate for the subsequent distance covered is Rs 4 per km. Find two solutions of the equation. What is the total fare if a boy has traveled 10 km?
Answer (2)
Find two solutions for the equation 3 x + 2 y + 8 = 0 by setting x = 0 and solving for y , and then setting y = 0 and solving for x .
Calculate the taxi fare for the first kilometer: Rs 5.
Calculate the taxi fare for the subsequent 9 km: 9 km * Rs 4/km = Rs 36.
Calculate the total taxi fare: Rs 5 + Rs 36 = Rs 41. The solutions are ( 0 , − 4 ) and ( − 3 8 , 0 ) , and the total taxi fare is 41 .
Explanation
Understanding the Problem We are given the equation 3 x + 2 y + 8 = 0 and asked to find two solutions. A solution is a pair of values ( x , y ) that satisfy the equation. We can find solutions by choosing a value for x and solving for y , or vice versa.
Finding the First Solution Let's find the first solution by setting x = 0 . Substituting x = 0 into the equation gives 3 ( 0 ) + 2 y + 8 = 0 , which simplifies to 2 y + 8 = 0 . Subtracting 8 from both sides gives 2 y = − 8 . Dividing both sides by 2 gives y = − 4 . So, the first solution is ( 0 , − 4 ) .
Finding the Second Solution Now, let's find the second solution by setting y = 0 . Substituting y = 0 into the equation gives 3 x + 2 ( 0 ) + 8 = 0 , which simplifies to 3 x + 8 = 0 . Subtracting 8 from both sides gives 3 x = − 8 . Dividing both sides by 3 gives x = − 3 8 . So, the second solution is ( − 3 8 , 0 ) .
Understanding the Taxi Fare Next, we need to calculate the total taxi fare for a 10 km journey. The fare is Rs 5 for the first kilometer and Rs 4 per kilometer for the subsequent distance.
Fare for the First Kilometer The fare for the first kilometer is Rs 5.
Calculating Subsequent Distance The subsequent distance is 10 km - 1 km = 9 km.
Calculating Fare for Subsequent Distance The fare for the subsequent 9 km is 9 km * Rs 4/km = Rs 36.
Calculating Total Taxi Fare The total taxi fare is Rs 5 + Rs 36 = Rs 41.
Final Answer Therefore, two solutions of the equation 3 x + 2 y + 8 = 0 are ( 0 , − 4 ) and ( − 3 8 , 0 ) , and the total taxi fare for a 10 km journey is Rs 41.
Examples
Imagine you are planning a taxi ride in a new city. Knowing the base fare and the per-kilometer charge helps you estimate the total cost. Similarly, understanding linear equations like 3 x + 2 y + 8 = 0 can help you model relationships between different variables, such as the cost of items and the quantity you can buy with a fixed budget. This problem combines both algebraic solutions and real-world cost calculations, providing a practical application of math in everyday scenarios.
The two solutions for the equation 3 x + 2 y + 8 = 0 are ( 0 , − 4 ) and ( − 3 8 , 0 ) . The total taxi fare for a 10 km journey is Rs 41. This fare comprises Rs 5 for the first kilometer and Rs 36 for the subsequent 9 kilometers.
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