Find the solution to the system of equations. Please show your work
2x - 4y + 5z = -33
4x - y = -5
-2x + 2y - 3z =19
To solve this system of equations, we can use the method of substitution.
Step 1: Solve one equation for one variable in terms of the other variables.
Let's solve the second equation for y:
4x - y = -5
y = 4x + 5
Step 2: Substitute the expression found in step 1 into the other two equations.
Now we can substitute y = 4x + 5 into the first and third equations:
2x - 4(4x + 5) + 5z = -33
-2x + 2(4x + 5) - 3z = 19
Simplify both equations:
2x - 16x - 20 + 5z = -33
-2x + 8x + 10 - 3z = 19
Combine like terms:
-14x + 5z = 13
6x - 3z = 9
Step 3: Solve one of the equations from step 2 for one variable in terms of the other variable.
Let's solve the second equation for x:
6x - 3z = 9
6x = 3z + 9
x = (3z + 9)/6
x = (1/2)z + 3/2
Step 4: Substitute the expression found in step 3 into the remaining equation from step 2.
Substitute x = (1/2)z + 3/2 into -14x + 5z = 13:
-14((1/2)z + 3/2) + 5z = 13
-7z - 21 + 5z = 13
-2z = 34
z = -17
Step 5: Substitute the value found in step 4 into the expression found in step 3 to solve for x.
Substitute z = -17 into x = (1/2)z + 3/2:
x = (1/2)(-17) + 3/2
x = -17/2 + 3/2
x = -14/2
x = -7
Step 6: Substitute the values found in steps 4 and 5 into the expression from step 1 to solve for y.
Substitute x = -7 and z = -17 into y = 4x + 5:
y = 4(-7) + 5
y = -28 + 5
y = -23
Therefore, the solution to the system of equations is x = -7, y = -23, and z = -17.