Question

1)3х+5у=16, 2х+3у=9 Решите систему уравнений

Answer 1

Узнаем x через первое уравнение:

3x = 16 - 5y

x = 16/3 - 5y/3

Подстановка:

2(16/3 - 5y/3) + 3y = 9

32/3 - 10y/3 + 3y = 9

y/3 = 32/3 - 9

y = 32 - 27

y = 5;

Узнаем x через второе уравнение:

2x + 3*5 = 9

2x = 9 - 15

2x = -6

x = -3;

Answer 2

To solve the system of equations:

1. 3x + 5y = 16
2. 2x + 3y = 9

There are several methods to solve this system, such as substitution, elimination, or using matrices. I will use the elimination method:

Step 1: Multiply both sides of the second equation by 5 to make the coefficients of y in both equations equal:

5(2x + 3y) = 5(9) 10x + 15y = 45

Step 2: Now, we have two equations with equal coefficients of y. We can eliminate y by subtracting the first equation from the second equation:

(10x + 15y) - (3x + 5y) = 45 - 16 10x + 15y - 3x - 5y = 29 7x + 10y = 29

Step 3: Now, we have a new equation:

7x + 10y = 29

Step 4: We can isolate y in terms of x from this equation:

10y = 29 - 7x y = (29 - 7x) / 10

Step 5: Substitute the value of y into the first equation (3x + 5y = 16):

3x + 5((29 - 7x) / 10) = 16

Step 6: Solve for x:

Multiply through by 10 to get rid of the fraction:

30x + 5(29 - 7x) = 160

Expand:

30x + 145 - 35x = 160

Combine like terms:

-5x + 145 = 160

Now, isolate x:

-5x = 160 - 145 -5x = 15

x = 15 / (-5) x = -3

Step 7: Now that we have the value of x, we can find y by substituting x back into one of the original equations. Let's use the first equation:

3x + 5y = 16

Substitute x = -3:

3(-3) + 5y = 16

-9 + 5y = 16

Add 9 to both sides:

5y = 16 + 9 5y = 25

y = 25 / 5 y = 5

Step 8: The solution to the system of equations is x = -3 and y = 5.