To complete the equation \(5x + 12 = x + 9\) so that it has no solution, we want to manipulate it so that it leads to a contradiction.
One way to do this is to make the two sides equal to a different constant. For example, we could rewrite the equation as:
\[ 5x + 12 = x + 9 + k \]
where \(k\) is a non-zero number.
Let’s say we choose \(k = 5\), so we modify it to:
\[ 5x + 12 = x + 9 + 5 \]
This simplifies to:
\[ 5x + 12 = x + 14 \]
Now, let's solve for \(x\):
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Subtract \(x\) from both sides: \[ 5x - x + 12 = 14 \] \[ 4x + 12 = 14 \]
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Subtract 12 from both sides: \[ 4x = 2 \]
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Divide by 4: \[ x = \frac{1}{2} \]
This still yields a solution. Let’s adjust our approach.
Instead, we can use:
\[ 5x + 12 = x + 10 \]
Solving this, we get:
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Rearranging gives: \[ 5x + 12 - x = 10 \] \[ 4x + 12 = 10 \]
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Subtracting 12 from both sides gives: \[ 4x = -2 \]
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Which results in: \[ x = -\frac{1}{2} \]
This again yields a solution.
Let’s permanently destroy the solution: by making both sides equal but not aligned.
Instead, let’s use: \[ 5x + 12 = x + 5 \]
Now:
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Rearranging gives: \[ 5x - x + 12 = 5 \] \[ 4x + 12 = 5 \]
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Subtracting 12 from both sides gives: \[ 4x = -7 \]
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Which results in: \[ x = -\frac{7}{4} \]
We are heading in circles.
To get an equation with no solution:
Use:
5x + 12 = 5 (any number false). Thus: \[ 5x + 12 = 5 \] immediately false as we expand \(5x + 12 = 5 -\) no number solves.
So equation: \[ 5x + 12 = 5 \]
Now as such, no value matches; hence LEGITIMATELY returns no solution.