Final answer:
To solve the system of equations, we will use the method of elimination to eliminate variables one at a time. This will result in a system of two equations with two variables, which can be solved by substitution or elimination. The solution to the system is x = -4.38, y = 1.34, and z = 7.21.
Step-by-step explanation:
To solve the simultaneous equations:
Equation 1: 6x - 4y + 3z = -10
Equation 2: 3x - 5y - 3z = -1
Equation 3: -5x + 6y - 3z = 5
Step 1: We can use the method of elimination to eliminate one variable at a time. Start by cancelling out the y variable. Multiply Equation 1 by 3 and Equation 2 by 6:
18x - 12y + 9z = -30
18x - 30y - 18z = -6
Step 2: Subtract Equation 2 from Equation 1:
18x - 12y + 9z - 18x + 30y + 18z = -30 - (-6)
42y + 27z = -24
Step 3: Multiply Equation 1 by 5 and Equation 3 by 6:
30x - 20y + 15z = -50
-30x + 36y - 18z = 30
Step 4: Add Equation 3 to Equation 1:
30x - 20y + 15z - 30x + 36y - 18z = -50 + 30
16y - 3z = -20
Step 5: Now we have a system of two equations with two variables:
42y + 27z = -24
16y - 3z = -20
Step 6: Solve the system using any method you prefer. For simplicity, let's use the method of elimination again. Multiply the second equation by 7:
112y - 21z = -140
42y + 27z = -24
Step 7: Subtract the second equation from the first:
112y - 21z - 42y - 27z = -140 - (-24)
70y - 48z = -116
Step 8: Now we have a system of two linear equations with two variables:
70y - 48z = -116
16y - 3z = -20
Step 9: Solve the system using any method you prefer. For simplicity, let's use the method of substitution. Solve Equation 2 for y:
y = (3z - 20)/16
Step 10: Substitute this value of y into Equation 1:
70(3z - 20)/16 - 48z = -116
210z - 1400 - 48z = -232C
162z - 1400 = -232
162z = -232 + 1400
162z =1168
z = 1168/162
z = 7.21
Step 11: Substitute the value of z into the equation for y:
y = (3(7.21) - 20)/16
y = 21.47/16
y = 1.34
Step 12: Substitute the values of y and z into any of the original equations to solve for x. Let's use Equation 1:
6x - 4(1.34) + 3(7.21) = -10
6x - 5.36 + 21.63 = -10
6x + 16.27 = -10
6x = -10 - 16.27
6x = -26.27
x = -26.27/6
x = -4.38
Step 13: Therefore, the solution to the system of equations is
x = -4.38,
y = 1.34, and
z = 7.21.