Solve by Substitution
x + 2y + z = 14
y = z + 1
x = -3z + 6
How do I find z (or any other variable for that matter)?
−2z + 6 + 2(z+1) = 14
−2z + 6 + 2z + 2 = 14
8 = 14
I try to solve the equation but end up with this, which is false.
To solve this system of equations using substitution, we will substitute the values of x and y into the first equation.
Given:
1) x + 2y + z = 14
2) y = z + 1
3) x = -3z + 6
We will start by substituting the value of x from equation 3) into equation 1):
-3z + 6 + 2y + z = 14
Simplify the equation:
-2z + 2y + 6 = 14
Next, we will substitute the value of y from equation 2) into the equation:
-2z + 2(z + 1) + 6 = 14
Simplify and solve for z:
-2z + 2z + 2 + 6 = 14
8 = 14
Since the equation is inconsistent, there is no solution to this system of equations.
To solve the given system of equations by substitution, we'll substitute the expressions for y and x into the first equation. Let's start with the equation y = z + 1.
1. First, substitute y in the first equation with its expression:
x + 2(y) + z = 14
x + 2(z + 1) + z = 14
2. Simplify the expression:
x + 2z + 2 + z = 14
x + 3z + 2 = 14
Now, let's substitute x in the first equation using its expression:
3. Substitute x in the simplified equation:
(-3z + 6) + 3z + 2 = 14
4. Simplify and solve for z:
-3z + 6 + 3z + 2 = 14
-3z + 3z + 8 = 14
8 = 14
Since this equation does not have a solution (8 ≠ 14), it means that the system of equations is inconsistent, and there is no solution. The given set of equations does not have a common solution for x, y, and z.
y = z+1 and x = -3z+6, so
(-3z+6)+2(z+1) + z = 14
so, find z, then x and y