To determine the missing values that would make the equation have zero solutions, we start by analyzing the given equation:
\[ 2(4x + 6) = 3x + 12 \]
Let's simplify both sides.
Left Side: \[ 2(4x + 6) = 8x + 12 \]
Right Side: We have \( 3x + 12\), which is already simplified.
Now our equation looks like: \[ 8x + 12 = 3x + 12 \]
Next, we'll move all the \(x\) terms to one side and the constant terms to the other side.
Subtract \(3x\) from both sides: \[ 8x - 3x + 12 = 12 \] This simplifies to: \[ 5x + 12 = 12 \]
Next, we subtract \(12\) from both sides: \[ 5x = 0 \]
Now, divide by \(5\): \[ x = 0 \]
This gives us one solution (i.e., \(x = 0\)). To make the equation have zero solutions, the left and right sides must be equal but not with any \(x\) term. This means we need a situation where the equation simplifies to a statement that is not true (like \(12 = 0\) or something like that).
To do this, we set up the situation as follows:
- Make the coefficients of \(x\) equal on both sides.
- Adjust the constant so that they do not equal each other.
For instance, we can adjust the constants such that:
Let’s equate the coefficients of \(x\) on both sides: Let’s keep the left side as \(2(4x + b)\) where \(b\) is some constant, and let’s say the right side is \(3x + c\).
From our previous observation, we can say: \[ 2(4x + b) = 3x + c \] Gives us: \[ 8x + 2b = 3x + c \]
Setting: \[ 8x - 3x = c - 2b \]
This leads to: \[ 5x = c - 2b \]
To make this have no solution:
- Set \(c = 2b + 1\) (you could also say \(c = 2b + 5\) or any variation of this as long as it does not equal \(0\))
- This would make our equation be \(5x = 1\) (or \(5x = 5\) etc.), which does not hold true, hence giving no solutions.
For example: Let’s take specific values: Let’s have \(2(4x + 1)\) on left side and \(3x + 3\) on the right. Then we have:
\[ 2(4x + 1) = 3x + 3 \] This results in \(8x + 2 = 3x + 3\), leading to: \[ 5x + 2 = 3 \] Substituting these values shows that when you operate on them, you do not get a true statement, leading to no solutions.
Lastly, substitute appropriate values to verify conditions for being inconsistent, and ensure these do not balance, leading to zero solutions. Adjust \(b\) and \(c\) accordingly to achieve these properties.