Answer:
To solve the system of equations:
3x + 2y - 5z = 3
2x - 4y + 6z = 2
x + 2y - z = 13
Explanation:
We can use any suitable method, such as substitution, elimination, or matrix methods. Here, we'll use the elimination method to find the solution.
Step 1: Let's eliminate one variable from equations (1) and (2). We can do this by multiplying equation (1) by 2 and adding it to equation (2):
2 * (3x + 2y - 5z) + (2x - 4y + 6z) = 2 * 3 + 2
6x + 4y - 10z + 2x - 4y + 6z = 8
8x - 4z = 8
Step 2: Now, let's eliminate another variable from equations (1) and (3). We can do this by multiplying equation (1) by 1 and adding it to equation (3):
1 * (3x + 2y - 5z) + (x + 2y - z) = 1 * 3 + 13
3x + 2y - 5z + x + 2y - z = 16
4x + 4y - 6z = 16
Step 3: Now, we have two new equations:
8x - 4z = 8
4x + 4y - 6z = 16
Step 4: Let's simplify equation (1) by dividing all terms by 4:
2x - z = 2
Step 5: Now, we have two equations:
2x - z = 2
4x + 4y - 6z = 16
Step 6: Let's solve equation (1) for z:
z = 2x - 2
Step 7: Substitute the value of z into equation (2):
4x + 4y - 6(2x - 2) = 16
4x + 4y - 12x + 12 = 16
-8x + 4y = 4
Step 8: Simplify equation (3) by dividing all terms by 4:
-2x + y = 1
Step 9: Now, we have two equations:
z = 2x - 2
-2x + y = 1
Step 10: We can now solve the above equations simultaneously. Let's solve equation (2) for y:
y = 1 + 2x
Step 11: Substitute the value of y into equation (1):
z = 2x - 2
So, the solution for the system is:
x = x (can take any real value)
y = 1 + 2x
z = 2x - 2
The solution is not unique; it depends on the value of x. Therefore, option D ("none") i