Final answer:
To solve the given system of equations, we can use the method of elimination. After performing the necessary steps, we find that the solution of the system is (x = 11, y = -5.5, z = -15.5).
Step-by-step explanation:
To solve the given system of equations:
3x + 4y + 3z = 5
2x + 2y + 3z = 5
5x + y + 3z = 3
We can use the method of elimination to find the values of x, y, and z.
1. Multiply the second equation by 2 and subtract it from the first equation:
(3x + 4y + 3z) - 2(2x + 2y + 3z) = 5 - 2(5)
x + 4y + 3z - 4x - 4y - 6z = 5 - 10
-x - 3z = -5
2. Multiply the third equation by 3 and subtract it from the first equation:
(3x + 4y + 3z) - 3(5x + y + 3z) = 5 - 3(3)
x + 4y + 3z - 15x - 3y - 9z = 5 - 9
-14x + y - 6z = -4
3. Multiply the third equation by 2 and subtract it from the second equation:
(2x + 2y + 3z) - 2(5x + y + 3z) = 5 - 2(3)
2x + 2y + 3z - 10x - 2y - 6z = 5 - 6
-8x - 3z = -1
4. Now we have a system of two equations:
-x - 3z = -5
-8x - 3z = -1
5. Add the two equations together:
(-x - 3z) + (-8x - 3z) = (-5) + (-1)
-9x - 6z = -6
6. Divide both sides of the equation by -3:
-9x - 6z = -6
3x + 2z = 2
7. Now substitute the value of z from the second equation into the first equation:
-x - 3z = -5
-x - 3(-2) = -5
-x + 6 = -5
-x = -5 - 6
-x = -11
x = 11
8. Substitute the values of x and z into the second equation:
3x + 2z = 2
3(11) + 2z = 2
33 + 2z = 2
2z = 2 - 33
2z = -31
z = -31/2
z = -15.5
9. Substitute the values of x and z into the third equation:
5x + y + 3z = 3
5(11) + y + 3(-15.5) = 3
55 + y - 46.5 = 3
y - 46.5 = 3 - 55
y - 46.5 = -52
y = -52 + 46.5
y = -5.5
Therefore, the solution to the system of equations is (x = 11, y = -5.5, z = -15.5).