Question

Решить систему уравнений 9x-6y-4z=5 15x-7y+5z=-10 10x+5y-7z=30

Answer 1

Решение методом Крамера на фото

Answer 2

To solve the system of equations:

1. 9x - 6y - 4z = 5
2. 15x - 7y + 5z = -10
3. 10x + 5y - 7z = 30

We can use the method of elimination or substitution to find the values of x, y, and z that satisfy all three equations.

Method 1: Elimination Step 1: Multiply the equations to create a common coefficient to eliminate a variable. Step 2: Add or subtract the equations to eliminate the variable. Step 3: Solve for the remaining variables.

Step 1: Let's multiply the equations to eliminate the variable "y." Multiply Equation 1 by 7 and Equation 2 by 6:

Equation 1: 63x - 42y - 28z = 35 Equation 2: 90x - 42y + 30z = -60

Step 2: Subtract Equation 1 from Equation 2 to eliminate "y":

(90x - 42y + 30z) - (63x - 42y - 28z) = -60 - 35

Simplify the equation:

90x - 42y + 30z - 63x + 42y + 28z = -95

27x + 58z = -95 (Equation 4 after elimination)

Step 3: Now let's eliminate "y" again by multiplying Equation 1 by 5 and Equation 3 by 6:

Equation 1: 45x - 30y - 20z = 25 Equation 3: 60x + 30y - 42z = 180

Add Equation 1 and Equation 3 to eliminate "y":

(45x - 30y - 20z) + (60x + 30y - 42z) = 25 + 180

Simplify the equation:

105x - 62z = 205 (Equation 5 after elimination)

Now we have a system of two equations with two variables:

1. 27x + 58z = -95 (Equation 4)
2. 105x - 62z = 205 (Equation 5)

Method 2: Substitution Step 1: Solve one equation for one variable and substitute it into the other equations. Step 2: Solve the resulting system of two equations with two variables.

Let's solve Equation 3 for "y":

10x + 5y - 7z = 30 5y = -10x - 30 + 7z y = (-2x - 6) + (7/5)z

Now substitute the expression for "y" into the first two equations:

1. 9x - 6((-2x - 6) + (7/5)z) - 4z = 5
2. 15x - 7((-2x - 6) + (7/5)z) + 5z = -10

Now simplify these equations:

1. 9x + 12x + 36 - 6(7/5)z - 4z = 5

2. 15x + 14x + 42 - 7(7/5)z + 5z = -10

3. (21x - (34/5)z = -31 (Equation 6)

4. (29x - (24/5)z = -52 (Equation 7)

Now we have a system of two equations with two variables:

1. 21x - (34/5)z = -31 (Equation 6)
2. 29x - (24/5)z = -52 (Equation 7)

Both methods yield the same system of equations:

1. 27x + 58z = -95 (Equation 4)
2. 105x - 62z = 205 (Equation 5)

Now, let's solve this system. One way to do it is to solve Equation 4 for x and then substitute the value of x into Equation 5:

Equation 4: 27x + 58z = -95 27x = -58z - 95 x = (-58z - 95)/27

Now substitute the value of x into Equation 5:

105((-58z - 95)/27) - 62z = 205

Now, solve for z:

105(-58z - 95) - 62z * 27 = 205 * 27 -6090z - 10075 - 1674z = 5535 -7764z = 15610 z = -15610 / 7764 z ≈ -2.0102

Now, substitute the value of z back into the expression for x:

x = (-58z - 95)/27 x = (-58(-2.0102) - 95)/27 x ≈ 3.9969

Finally, find the value of y using any of the original equations:

10x + 5y - 7z = 30 10(3.9969) + 5y - 7(-2.0102) = 30 39.969 + 5y + 14.0714 = 30 5y = 30 - 39.969 - 14.0714 5y = -23.0404 y = -23.0404 / 5 y ≈ -4.6081

So, the solution to the system of equations is approximately: x ≈ 3.9969 y ≈ -4.6081 z ≈ -2.0102