Solve by the system of equations by substitution.
x + 2y + z = 14
y = z + 1
x = -3z + 6
Anyone able to help me with this? I've been stuck on it all day and can't seem to solve it, it doesn't make any sense to me.
well, geez - start substituting!
y=z+1
x=-3z+6, so
(-3z+6)+2(z+1)+z=14
Now just solve for z, then you can get x and y.
Certainly! I'll guide you through the process of solving this system of equations by substitution.
Step 1: Start by solving one equation for one variable in terms of the other variables. Let's solve the second equation, y = z + 1, for y.
Step 2: Substitute the expression we found for y (z + 1) into the first equation, x + 2y + z = 14. Replace y with z + 1, so the equation becomes:
x + 2(z + 1) + z = 14.
Step 3: Simplify and solve this equation for x in terms of z. Distribute the 2:
x + 2z + 2 + z = 14
x + 3z + 2 = 14
x = 12 - 3z.
Step 4: Substitute the expressions we found for x and y into the third equation, x = -3z + 6. Replace x with 12 - 3z, so the equation becomes:
12 - 3z = -3z + 6.
Step 5: Simplify and solve this equation for z. Move all the terms involving z to one side of the equation:
-3z + 3z = 6 - 12
0 = -6.
Step 6: Analyze the result. We have obtained an equation that is always satisfied, 0 = -6. This indicates that the system of equations is inconsistent, meaning there is no solution that satisfies all three equations simultaneously.
In summary, the system of equations you provided has no solution.