Answer:
This equation contains two variables, x and z. We need one more equation to solve for both x and z. However, without an additional equation, we cannot determine unique values for x, y, and z.
It is important to note that the original system of equations may be inconsistent or dependent, which means there may not be a unique solution. To obtain a unique solution, we would need another equation to create a consistent and independent system.
Explanation:
To solve the system of equations:
10x + 7y - 2z = 46
3x - 2y + 9z = 22
5x + y - 3z = 28
We can use the method of elimination or substitution to find the values of x, y, and z that satisfy all three equations simultaneously. Let's use the method of elimination.
First, we can eliminate the variable x by multiplying the second equation by 10 and the third equation by -5:
10(3x - 2y + 9z) = 10(22) [Multiply the second equation by 10]
-5(5x + y - 3z) = -5(28) [Multiply the third equation by -5]
This simplifies to:
30x - 20y + 90z = 220
-25x - 5y + 15z = -140
Next, we can add the resulting equations to eliminate the variable x:
(30x - 20y + 90z) + (-25x - 5y + 15z) = 220 + (-140)
This simplifies to:
5x - 25y + 105z = 80
Now, we have two equations:
5x - 25y + 105z = 80 [Resulting equation from eliminating x]
10x + 7y - 2z = 46 [Original first equation]
To eliminate the variable x again, we can multiply the second equation by -2:
-2(10x + 7y - 2z) = -2(46)
This simplifies to:
-20x - 14y + 4z = -92
Now, we can add the resulting equation to the first equation:
(5x - 25y + 105z) + (-20x - 14y + 4z) = 80 + (-92)
This simplifies to:
-15x - 39y + 109z = -12
We now have two equations:
-15x - 39y + 109z = -12 [Resulting equation from eliminating x]
10x + 7y - 2z = 46 [Original first equation]
To eliminate the variable y, we can multiply the second equation by 39:
39(10x + 7y - 2z) = 39(46)
This simplifies to:
390x + 273y - 78z = 1794
Now, we can add the resulting equation to the first equation:
(-15x - 39y + 109z) + (390x + 273y - 78z) = -12 + 1794
This simplifies to:
375x + 234z = 1782
We now have one equation:
375x + 234z = 1782