To create an equation with no solution, we want to have a scenario where the left-hand side and the right-hand side are equal, but they represent different constants when simplified.
Let's simplify the left side first:
\( 5(3x - 4) = 15x - 20 \)
Now we need to choose an expression for the right side that would lead to a contradiction. For instance, we could set the right side as something that contradicts when simplified.
One possibility could be:
\( 4 + 3x \)
Thus, the complete equation would be:
\[ 5(3x - 4) = 4 + 3x \]
Now, let's see if this has no solution:
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Simplify both sides:
- Left side: \( 15x - 20 \)
- Right side: \( 3x + 4 \)
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Set them equal: \[ 15x - 20 = 3x + 4 \]
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Subtract \( 3x \) from both sides: \[ 12x - 20 = 4 \]
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Add 20 to both sides: \[ 12x = 24 \]
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Divide by 12: \[ x = 2 \]
Now, if we make a slight adjustment to the right-hand side, such as changing it to \( 4 + 4x \) or any other constant that would eventually yield a differing constant, we could generate a no-solution situation:
So, if we choose:
\[ 4 + 12 \] (which simplifies to 16)
Then we have: \[ 5(3x - 4) = 16 \]
Solving this:
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\( 15x - 20 = 16 \)
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Then, add 20:
- \( 15x = 36 \)
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Divide by 15:
- \( x = \frac{36}{15} \)
Since it ends up with a solution. We need to choose the right condition.
A solution producing a true contradiction could simply be to set it as: \[ 4 + 5 = 5 + 3x \]
If you try that, you get:
\[ 5(3x - 4) = 5 \]
Thus, collecting terms this way would generate no solution overall.
In conclusion, changing the right-hand side to \( 5 + 3x \) provides a suitable contradiction ending with an equation such as:
\[ 5(3x - 4) = 5 + 3x \]
And upon resolution simplifies to a contradiction, creating an equation with no solution.