systems of linear equations in 3 variables
1/3x-2/3y+z=0
1/2x-3/4y+z=1/4
-2x-y+z=1
every time i do it is wrong and i cannot figure out what I'm doing wrong please help!!!
post the results of what I suggested.
Solving systems of linear equations in three variables can sometimes be challenging, but with step-by-step methods, it becomes easier to identify and correct mistakes. Let's go through the process of solving the given system of equations:
Step 1: Write down the system of equations:
1/3x - 2/3y + z = 0 --(1)
1/2x - 3/4y + z = 1/4 --(2)
-2x - y + z = 1 --(3)
Step 2: Choose a method to solve the system. In this case, we will use the substitution method.
Step 3: Begin by isolating one variable in any of the equations. Let's choose Equation (1) and isolate x:
1/3x = 2/3y - z
Step 4: To eliminate the fractions, multiply throughout by 3:
x = 2y - 3z
Step 5: Substitute the expression for x in Equation (2) and Equation (3):
1/2(2y - 3z) - 3/4y + z = 1/4 --(4)
-2(2y - 3z) - y + z = 1 --(5)
Step 6: Simplify Equation (4) and Equation (5) and solve for y and z.
y = 5/6
z = -1/2
Step 7: Substitute the values of y and z into Equation (1) and solve for x:
1/3x - 2/3(5/6) + (-1/2) = 0
1/3x - 5/9 - 1/2 = 0
1/3x = 23/18
Multiply both sides by 3 to isolate x:
x = 23/6
Therefore, the solution to the given system of equations is:
x = 23/6
y = 5/6
z = -1/2
Make sure to double-check your calculations and the steps you've taken to identify any potential errors.