Let's analyze each equation systematically to determine the type of solution they have.
- Equation: \( 3z + 12 + 2 = 8 + 4 + z \)
First, simplify both sides:
Left-hand side: \( 3z + 12 + 2 = 3z + 14 \)
Right-hand side: \( 8 + 4 + z = 12 + z \)
Now the equation looks like: \[ 3z + 14 = 12 + z \]
Next, let's isolate \( z \). Subtract \( z \) from both sides: \[ 3z - z + 14 = 12 \] \[ 2z + 14 = 12 \]
Now, subtract 14 from both sides: \[ 2z = 12 - 14 \] \[ 2z = -2 \]
Finally, divide by 2: \[ z = -1 \]
Since we found a specific value for \( z\), this means this equation has one solution.
- Equation: \( 4(z + 1) = 4z + 1 \)
First, expand the left-hand side: \[ 4(z + 1) = 4z + 4 \]
So now the equation looks like: \[ 4z + 4 = 4z + 1 \]
Next, we subtract \( 4z \) from both sides: \[ 4 = 1 \]
This results in a false statement. Since there is no value of \( z \) that can make \( 4 = 1 \) true, this equation has no solution.
Summary:
- Item 1: One solution because when you solve for \( z \), the result is \( z = -1 \).
- Item 2: No solution because you end up with a false statement - no value of \( z \) will make the equation true.
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