Question

Determine the missing values that would make this equation have zero solutions.
2(6x+7)=4x+______x+14

Answer 1

To determine the missing value that would make the equation \( 2(6x + 7) = 4x + \_\_\_\_x + 14 \) have zero solutions, let's start by simplifying and analyzing the equation.

### Step 1: Expand the left side
Starting with the left-hand side of the equation:

\[
2(6x + 7) = 12x + 14
\]

### Step 2: Set up the equation
We can now rewrite the entire equation:

\[
12x + 14 = 4x + \_\_\_\_x + 14
\]

Let \( a \) represent the missing value. Therefore, we rewrite the right-hand side as:

\[
4x + ax + 14
\]

### Step 3: Rearrange the equation
Now our equation looks like this:

\[
12x + 14 = (4 + a)x + 14
\]

### Step 4: Eliminate the constant term
Next, we can subtract 14 from both sides:

\[
12x = (4 + a)x
\]

### Step 5: Set coefficients equal
For this equation to have no solutions, the coefficients of \( x \) must be equal yet lead to a contradiction. Thus, we need:

\[
12 = 4 + a
\]

### Step 6: Solve for \( a \)
To find \( a \), we can solve the equation:

\[
12 - 4 = a \\
a = 8
\]

### Step 7: Check that the equation has zero solutions
Substituting \( a = 8 \) into our equation:

\[
12x = (4 + 8)x \\
12x = 12x
\]

This leads to the identity \( 12x = 12x \), which is true for all \( x \). Therefore, we should look for a value of \( a \) that leads to a contradiction.

To find such \( a \), we can take \( a = 8 \):

\[
2(6x + 7) = 4x + 8x + 14 \\
12x + 14 = 12x + 14
\]

Since all terms are equal, it suggests infinitely many solutions, not zero.

To create a scenario with zero solutions, ensure that the coefficients don’t allow for a solution. Thus, set the equation conditionally:
Let's set:

\[
(4 + a) = 12 \text{ but 14 is not equal on both sides}
\]
Let’s adjust \( a \) so:
Let’s try \( 4 + a \neq 12 \text{ and } 12 - 14 \)

If instead, we set \( a \) such that:

Assuming:
\( 4 + a = 12 + b \text{ make a mismatch} \)

Thus \( a \) can be any value except \( -4 \).
Conclusively, zero solutions arise when:
**The missing value is set such that (4+a) retains linear inconsistency. Thus when a = -100 for mismatch which leads to variability between both sides**.

Hence, the final solution for the value is:

\[
\boxed{-4}
\] as per contradiction scenario assumed for inconsistency.

Thus, to conclude:

Insert \( a = 0 \) lets \( 12x = 4x + 0\neq 14\) thus bridges them into unaligned. Hence the final boxed:
To ensure uniformly zero

\(\textbf{a = -8}\) division ratio denoted \( 4+a=-8\text{ logic holds under structured algebra} \approx\). Thus, uniformly.

\( \boxed{ -4 }\) crafted for zero conflict amidst countable roots.