Solving the System of Equations
Let’s solve the given system of equations step-by-step:
- Equation 1: 2x + 3y + z = 1
- Equation 2: 3x + y + 2z = 12
- Equation 3: x + 2y + 3 = 5
Step 1: Simplify Equation 3
First, let’s simplify Equation 3:
- x + 2y = 5 – 3
- x + 2y = 2
Step 2: Express x in terms of y
From the simplified Equation 3, we can express x in terms of y:
- x = 2 – 2y
Step 3: Substitute x into Equation 1 and Equation 2
Next, we’ll substitute x = 2 – 2y into Equations 1 and 2.
Substituting into Equation 1:
- 2(2 – 2y) + 3y + z = 1
- 4 – 4y + 3y + z = 1
- -y + z = 1 – 4
- z = y – 3
Substituting into Equation 2:
- 3(2 – 2y) + y + 2z = 12
- 6 – 6y + y + 2z = 12
- -5y + 2z = 12 – 6
- 2z = 6 + 5y
- z = 3 + (5/2)y
Step 4: Set the two equations for z equal to each other
We have two expressions for z:
- z = y – 3
- z = 3 + (5/2)y
Now we can set them equal:
- y – 3 = 3 + (5/2)y
- y – (5/2)y = 3 + 3
- -3/2y = 6
- y = -4
Step 5: Substitute y back to find x and z
Now, substitute y = -4 back into the expressions for x and z:
- x: x = 2 – 2(-4) = 2 + 8 = 10
- z: z = -4 – 3 = -7
Final Solution
The solution to the system of equations is:
- x = 10
- y = -4
- z = -7