x + y = 4 , x + z = 4, y + z = 4
1st - 2nd: y-z=0
y = z
2nd - 3rd: x-y=0
x = y
so x = y = z
so from 1st : 2x = 4
x = 2
so x = y = z = 2
x+y + 0z=4
x+0y+z=4
0x+y+z=4
1,1,0,4
1,0,1,4
0,1,1,4
http://www.gregthatcher.net//Mathematics/GaussJordan.aspx
all variables are equal to 2
To find the values of x, y, and z that satisfy the three equations:
1. x + y = 4
2. x + z = 4
3. y + z = 4
We can use a method called substitution. Here's how you can solve it:
Step 1: Solve one of the equations for one variable in terms of the other.
Let's solve equation 1 for x:
x = 4 - y
Step 2: Substitute the value of x in the other two equations.
Substitute (4 - y) for x in equation 2:
4 - y + z = 4
Substitute (4 - y) for x in equation 3:
y + z = 4
Step 3: Simplify each equation.
Equation 2 becomes:
-z + y = 0
Equation 3 remains the same:
y + z = 4
Step 4: Combine the two equations into one equation.
Add equation 2 and equation 3 together:
(-z + y) + (y + z) = 0 + 4
Simplifying the equation:
2y = 4
Step 5: Solve the equation for y.
Divide both sides of the equation by 2:
y = 2
Step 6: Substitute the value of y into one of the original equations to solve for x or z.
Let's use equation 1:
x + 2 = 4
x = 2
Step 7: Substitute the values of x and y back into the remaining equation to solve for z.
Using equation 3:
2 + z = 4
z = 2
So the solution to the system of equations is:
x = 2, y = 2, z = 2