Now, we can solve this system of equations by adding or subtracting these equations to eliminate a variable:
Multiplying equation (4) by 5 and equation (5) by 7:
35x - 25y + 40z = 120 35x - 56y + 56z = 182
Subtracting these two equations:
31y - 16z = -62
Now, we can solve for one variable and substitute back to find the values of the other variables. Let's solve for z:
16z = 31y + 62 z = (31y + 62) / 16
Now, we can substitute this expression back into one of the equations to find the values of x and y. Let's substitute it into equation (4) for example:
To find the solution to this system of equations, we can use the method of substitution or elimination.
Let's use the elimination method to solve this system of equations:
4x - 4y + 3z = 16 (1)
3x - y + 5z = 8 (2)
2x - 7y + 3z = 18 (3)
Let's start by adding equations (1) and (2) to eliminate y:
4x - 4y + 3z + 3x - y + 5z = 16 + 8
7x - 5y + 8z = 24 (4)
Now, we can add equations (2) and (3) to eliminate y:
3x - y + 5z + 2x - 7y + 3z = 8 + 18
5x - 8y + 8z = 26 (5)
Now, we have two new equations:
7x - 5y + 8z = 24 (4)
5x - 8y + 8z = 26 (5)
Now, we can solve this system of equations by adding or subtracting these equations to eliminate a variable:
Multiplying equation (4) by 5 and equation (5) by 7:
35x - 25y + 40z = 120
35x - 56y + 56z = 182
Subtracting these two equations:
31y - 16z = -62
Now, we can solve for one variable and substitute back to find the values of the other variables. Let's solve for z:
16z = 31y + 62
z = (31y + 62) / 16
Now, we can substitute this expression back into one of the equations to find the values of x and y. Let's substitute it into equation (4) for example:
7x - 5y + 8(31y + 62) / 16 = 24
7x - 5y + 124y + 248 = 24
7x + 119y = -224
7x = -119y - 224
x = (-119y - 224)/7
Now we have expressions for x, y, and z in terms of y. Further simplification would be needed to find the exact values of x, y and z.