Question

Составьте систему уравнений и решите задачу :за 3 часа против течения лодка проплыла на 5 км больше ,чем за 2ч по течению. Скорость лодки против течения составляет 0,75 её скорости по течению. Какое расстояние прошла лодка за это время?

Answer 1

скорость лодки по течению х, расстояние по течению у
2*х=у  по течению
3*(0,75х)=у+5  против течения

2х=у
2,25х=у+5

2,25х=2х+5
0,25х=5
х=20  скорость лодки по течению

2*20=40 км по течению
 40+5=45км против течения

Answer 2

Problem Analysis

To solve this problem, we need to set up a system of equations based on the given information and then solve for the distance traveled by the boat.

Let's denote the speed of the boat in still water as x km/h and the speed of the current as y km/h.

According to the problem, the boat traveled 5 km more in 3 hours against the current than in 2 hours with the current. We can set up the following equation based on this information:

3(x - y) = 2(x + y) + 5 We also know that the speed of the boat against the current is 0.75 times its speed with the current. We can set up another equation based on this information:

x - y = 0.75(x + y) We can solve this system of equations to find the values of x and y and then calculate the distance traveled by the boat in 3 hours.

Solving the System of Equations

Let's solve the system of equations and to find the values of x and y.

From equation we can simplify it as follows:

x - y = 0.75x + 0.75y

Rearranging the equation:

0.25x - 1.75y = 0 Now, let's substitute the value of x - y from equation into equation:

3(0.25x - 1.75y) = 2(x + y) + 5

Simplifying the equation:

0.75x - 5.25y = 2x + 2y + 5 Rearranging the equation:

0.75x - 2x = 5.25y + 2y + 5

-1.25x = 7.25y + 5 Now, we have two equations: and Let's solve them simultaneously.

Multiplying equation by 1.25 to eliminate the coefficient of x:

1.25(0.25x - 1.75y) = 1.25(0)

Simplifying the equation:

0.3125x - 2.1875y = 0 Adding equation to equation:

0.3125x - 2.1875y + (-1.25x) = 0 + 7.25y + 5

Simplifying the equation:

-0.9375x - 9.4375y = 5 Now, we have two equations: and Let's solve them simultaneously.

Multiplying equation by 4 to eliminate the coefficient of y:

4(-0.9375x - 9.4375y) = 4(5)

Simplifying the equation:

-3.75x - 37.75y = 20 Adding equation to equation:

-3.75x - 37.75y + (-1.25x) = 20 + 7.25y + 5

Simplifying the equation:

-5x - 45y = 25 Now, we have two equations: and Let's solve them simultaneously.

Multiplying equation by -1 to eliminate the coefficient of x:

-1(-5x - 45y) = -1(25)

Simplifying the equation:

5x + 45y = -25 Adding equation to equation:

5x + 45y + (-5x - 45y) = -25 + 25

Simplifying the equation:

0 = 0

The equation 0 = 0 indicates that the two equations are dependent, meaning they represent the same line. This means that there are infinitely many solutions to the system of equations.

We can choose any value for x or y and solve for the other variable. Let's choose x = 1.

From equation we can solve for y:

1 - y = 0.75(1 + y)

Simplifying the equation:

1 - y = 0.75 + 0.75y

Rearranging the equation:

1 - 0.75 = 0.75y + y

0.25 = 1.75y

Dividing both sides by 1.75:

y = 0.25 / 1.75

y = 0.1429

Now, we have the values of x = 1 and y = 0.1429.

Calculating the Distance Traveled

We can now calculate the distance traveled by the boat in 3 hours using the value of x.

The distance traveled by the boat in 3 hours against the current is:

3(x - y) = 3(1 - 0.1429) = 3(0.8571) = 2.5713 km

Therefore, the boat traveled approximately 2.5713 km in 3 hours.

Note: The solution to this problem may vary depending on the chosen values for x or y.