okay heres one of the ones i need help on can someone explain it to me thanks.
Solve each system of equations.
Let "z" be the parameter.
x +2y +3z =11
2x -y + z = 2
To solve this system of equations, we will use the method of substitution or elimination to find the values of x, y, and z.
Method 1: Substitution
Step 1: Solve one equation for one variable in terms of the other variables.
Let's solve the second equation for x:
2x - y + z = 2
2x = y - z + 2
x = (y - z + 2)/2
Step 2: Substitute the expression for x into the other equation.
x + 2y + 3z = 11
((y - z + 2)/2) + 2y + 3z = 11
Simplify the equation:
(y - z + 2) + 4y + 6z = 22
5y + 5z = 20
y + z = 4 (dividing both sides by 5)
Step 3: Solve for one variable. Let's solve for y:
y = 4 - z
Step 4: Substitute this value back into one of the original equations.
x = (y - z + 2)/2
x = (4 - z - z + 2)/2
x = (6 - 2z)/2
x = 3 - z
So, the solution to the system of equations is:
x = 3 - z
y = 4 - z
z = z (let z be the parameter)
Method 2: Elimination
Step 1: Multiply the first equation by 2 to make the coefficient of x in both equations equal.
2(x + 2y + 3z) = 2(11)
2x + 4y + 6z = 22
Now, we have:
2x + 4y + 6z = 22
2x - y + z = 2
Step 2: Subtract the second equation from the first equation to eliminate x.
(2x + 4y + 6z) - (2x - y + z) = 22 - 2
5y + 5z = 20
y + z = 4 (dividing both sides by 5)
Step 3: Solve for one variable. Let's solve for y:
y = 4 - z
Step 4: Substitute this value back into one of the original equations.
x + 2(4 - z) + 3z = 11
x + 8 - 2z + 3z = 11
Simplify the equation:
x + z = 3
So, the solution to the system of equations is:
x + z = 3
y = 4 - z
z = z (let z be the parameter)
In both methods, we end up with the same solution:
x + z = 3
y = 4 - z
z = z (let z be the parameter)
This means that the values of x, y, and z can take on different values based on the parameter z.