How can I solve the system of equations: 2x + y + z = 3, 2x + 2y + 3z = 2, and 3x + 3y + z = 4?

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To solve the given system of equations, we have the following equations:

  • 1. 2x + y + z = 3
  • 2. 2x + 2y + 3z = 2
  • 3. 3x + 3y + z = 4

Let’s solve the system step by step:

Step 1: Simplify the equations

The first equation remains as it is:

  • Equation (1): 2x + y + z = 3

From the second equation, we can simplify by dividing everything by 2:

  • Equation (2): x + y + 1.5z = 1

The third equation can also be simplified by dividing everything by 3:

  • Equation (3): x + y + 0.33z = rac{4}{3} ≈ 1.33

Step 2: Express variables in terms of others

From Equation (1), we can express z in terms of x and y:

  • z = 3 – 2x – y

Step 3: Substitute z into the other equations

Substituting z into Equation (2):

  x + y + 1.5(3 - 2x - y) = 1
  x + y + 4.5 - 3x - 1.5y = 1
  -2x - 0.5y + 4.5 = 1
  -2x - 0.5y = -3.5
  2x + 0.5y = 3.5
  4x + y = 7     (Equation 4)

Now, substituting z into Equation (3):

  x + y + 0.33(3 - 2x - y) = 1.33
  x + y + 0.99 - 0.66x - 0.33y = 1.33
  0.34x + 0.67y = 0.34
  x + 2y = 1 (Equation 5)

Step 4: Solve Equations 4 and 5

Now we have a new system to solve:

  • Equation (4): 4x + y = 7
  • Equation (5): x + 2y = 1

From Equation (5), we can express y in terms of x:

  y = (1 - x)/2

Substituting y into Equation (4):

  4x + (1 - x)/2 = 7
  8x + 1 - x = 14
  7x = 13
  x = rac{13}{7} ≈ 1.857

Now substituting back to find y:

  y = (1 - 13/7)/2 = (-6/7)/2 = -3/7

Finally, substituting values of x and y back into the equation for z:

  z = 3 - 2(13/7) - (-3/7) = 3 - 26/7 + 3/7 = 3 - 23/7 = rac{21 - 23}{7} = -rac{2}{7}

Final Solution

The solution to the system of equations is:

  • x = 13/7
  • y = -3/7
  • z = -2/7

Thus, the values of the variables that satisfy the system of equations are approximately:

  • x ≈ 1.857
  • y ≈ -0.429
  • z ≈ -0.286

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