Answer:
The solution of the system is (-2 , -1 , -3)
Explanation:
* In this system of equations we have three variables
* Equation (1) ⇒ 3x + 2y + z = 7
* Equation (2) ⇒ 5x + 5y + 4z = 3
* Equation (3) ⇒ 3x + 2y + 3z = 1
- Lets use the elimination method to solve
- Equation (1) and (equation (3) have same coefficients of x and y,
then we can start by subtracting equation (1) from equation (3) to
eliminate x and y and find z
∵ 3x + 2y + z = 7 ⇒ (1)
∵ 3x + 2y + 3z = 1 ⇒ (3)
- Subtract (1) from (3)
∴ (3x - 3x ) + (2y - 2y) + (3z - z) = (1 - 7)
∴ 2z = -6
- Divide both sides by 2
∴ z = -3
- Substitute the value of z in equations (1) and (2)
∵ 3x + 2y + (-3) = 1
∴ 3x + 2y - 3 = 1
- Add 3 for both sides
∴ 3x + 2y = 4 ⇒ (4)
∵ 5x + 5y + 4(-3) = 3
∴ 5x + 5y - 12 = 3
- Add 12 to both sides
∴ 5x + 5y = 15 ⇒ (5)
- Now we have system of equations of two variables
∵ 3x + 2y = 4 ⇒ (4)
∵ 5x + 5y = 15 ⇒ (5)
- Multiply equation (4) by -5 and equation (5) by 2 to eliminate y
∵ -5(3x) + -5(2y) = -5(4)
∵ 2(5x) + 2(5y) = 2(15)
∵ -15x - 10y = -20 ⇒ (6)
∵ 10x + 10y = 30 ⇒ (7)
- Add equations (6) and (7)
∴ -5x = 10
- Divide both sides by -5
∴ x = -2
- Substitute the value of x in equation (4) or (5) to find y
∵ 3(2) + 2y = 4
∴ 6 + 2y = 4
- Subtract 6 from both sides
∴ 2y = -2
- Divide both sides by 2
∴ y = -1
* The solution of the system is (-2 , -1 , -3)