To create an equation that has no solution, we can manipulate the original equation so that the two sides are equal but cannot be both true for any value of \(x\).
Starting with the original equation:
\[ -7x - 12 = x + 12 \]
Let's solve this equation:
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Add \(7x\) to both sides: \[ -12 = 8x + 12 \]
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Subtract \(12\) from both sides: \[ -12 - 12 = 8x \] \[ -24 = 8x \]
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Divide both sides by \(8\): \[ x = -3 \]
This shows that the original equation has a solution \(x = -3\).
Now, to make it have no solution, we can change it to something contradictory. Here’s one way to do it:
Change the original equation to:
\[ -7x - 12 = x + 15 \]
Now, let's solve this modified equation to see if it has a solution:
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Add \(7x\) to both sides: \[ -12 = 8x + 15 \]
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Subtract \(15\) from both sides: \[ -12 - 15 = 8x \] \[ -27 = 8x \]
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Divide both sides by \(8\): \[ x = -\frac{27}{8} \]
This still gives a solution.
Let’s instead adjust it further to:
\[ -7x - 12 = -7x + 10 \]
Now solving this:
- Add \(7x\) to both sides: \[ -12 = 10 \]
This is a contradiction (since \(-12\) does not equal \(10\)), indicating that the equation has no solution.
So the final equation is:
\[ -7x - 12 = -7x + 10 \]
This equation has no solution.