To solve the system of equations, we need to express it in a more manageable form:
- Equation 1: x + 2y = 6
- Equation 2: z + 3y + 2z = 7 simplifies to 3z + 3y = 7
- Equation 3: 3x + 2y + 5z = 4
Next, from Equation 1, we can express x in terms of y:
x = 6 – 2y
Now, substituting this expression for x into Equation 3:
3(6 – 2y) + 2y + 5z = 4
This expands to:
18 – 6y + 2y + 5z = 4
Combining like terms gives:
18 – 4y + 5z = 4
Rearranging results in:
5z = 4y – 14
Now, expressing z in terms of y gives:
z = (4y – 14) / 5
Next, substituting z back into the second equation:
3((4y – 14) / 5) + 3y = 7
Multiplying through by 5 to eliminate the fraction yields:
3(4y – 14) + 15y = 35
Expanding gives:
12y – 42 + 15y = 35
Combining like terms results in:
27y – 42 = 35
Adding 42 to both sides gives:
27y = 77
Finally, solving for y results in:
y = 77 / 27
Now substituting this value for y back into the equations will give us the values for x and z:
1. Substitute y into x = 6 – 2y
x = 6 – 2(77 / 27) = 6 – (154 / 27) = (162 – 154) / 27 = 8 / 27
2. Substitute y back into z = (4y – 14) / 5
z = (4(77 / 27) – 14) / 5 = (308 / 27 – 378 / 27) / 5 = (-70 / 27) / 5 = -14 / 27
Thus, the solution to our equations is:
x = 8/27, y = 77/27, z = -14/27