3x + 2y + z = 4
2x + y - 3z = 2
7x + 4y -5z = 8
How to solve?
one way is Gaussian elimination, as shown here:
http://www.gregthatcher.com/Mathematics/GaussJordan.aspx
Just enter your matrix of coefficients
To solve this system of equations, we can use the method of elimination or substitution. I will explain both methods, and you can choose the one that is more comfortable for you.
Method 1: Elimination
Step 1: Choose two equations and eliminate one variable. The goal is to eliminate the same variable from both equations.
Let's eliminate the variable 'x' from equations (1) and (2):
Multiply equation (1) by 2 and equation (2) by 3 to make the coefficients of 'x' the same:
2*(3x + 2y + z) = 2*(4) becomes 6x + 4y + 2z = 8.
3*(2x + y - 3z) = 3*(2) becomes 6x + 3y - 9z = 6.
Step 2: Subtract one equation from the other to eliminate the 'x' variable.
Subtract equation (2) from equation (1):
(6x + 4y + 2z) - (6x + 3y - 9z) = 8 - 6
Simplifying, we get: x + 7z = 2y + 2
Step 3: Repeat the process to eliminate another variable.
Now let's eliminate the variable 'x' from equations (1) and (3):
Multiply equation (1) by 7 and equation (3) by 3 to make the coefficients of 'x' the same:
7*(3x + 2y + z) = 7*(4) becomes 21x + 14y + 7z = 28.
3*(7x + 4y - 5z) = 3*(8) becomes 21x + 12y - 15z = 24.
Step 4: Subtract one equation from the other to eliminate the 'x' variable.
Subtract equation (3) from equation (1):
(21x + 14y + 7z) - (21x + 12y - 15z) = 28 - 24
Simplifying, we get: 2y + 22z = 4
Step 5: We now have a system of two equations in two variables:
x + 7z = 2y + 2 --------- (4)
2y + 22z = 4 --------- (5)
Now, we can solve this system using any method you prefer, such as substitution or elimination.
Method 2: Substitution
Step 1: Choose one equation and solve for one variable in terms of the other variables.
Let's solve equation (1) for 'x':
3x = 4 - 2y - z
Dividing throughout by 3, we get:
x = (4 - 2y - z) / 3
Step 2: Substitute this expression for 'x' in the other two equations.
Substituting in equation (2):
2*((4 - 2y - z)/3) + y - 3z = 2
Multiplying both sides by 3 to clear the fraction, we get:
4 - 2y - z + 3y - 9z = 6
Simplifying, we get:
y - 10z = 2 --------- (6)
Substituting in equation (3):
7*((4 - 2y - z)/3) + 4y - 5z = 8
Multiplying both sides by 3, we get:
28 - 14y - 7z + 12y - 15z = 24
Simplifying, we get:
-2y - 22z = -4 --------- (7)
Now, we have two equations with two variables:
y - 10z = 2 --------- (6)
-2y - 22z = -4 --------- (7)
Again, you can now solve this system using the method of your choice, such as substitution or elimination.