Let's analyze the problem step by step to verify if the student's solution is correct and to identify whether the original equation has a solution.
The original equation given is:
\[ 4(2 - 3x) = x - 2(2x + 1) \]
Expanding both sides:
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Left Side: \[ 4(2 - 3x) = 8 - 12x \]
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Right Side: \[ x - 2(2x + 1) = x - (4x + 2) = x - 4x - 2 = -3x - 2 \]
Now we have:
\[ 8 - 12x = -3x - 2 \]
Next, let's isolate \(x\). We can start by adding \(12x\) to both sides:
\[ 8 = 12x - 3x - 2 \]
This simplifies to:
\[ 8 = 9x - 2 \]
Now, add 2 to both sides:
\[ 10 = 9x \]
Dividing both sides by 9 gives:
\[ x = \frac{10}{9} \]
This means that the original equation does have a solution: \(x = \frac{10}{9}\).
Regarding the student's claim of "No solution," it appears they made an error in their calculations. Specifically, the step where they equated \(8 - 3x\) to \(-3x - 2\) was not followed through correctly in subsequent steps.
In conclusion, the original equation does have a solution:
The equation solved correctly would show that it has 1 solution.
1 answer