To determine the number of solutions for each equation, we can simplify and solve them one by one.
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Equation: \(3x - 5 = 3(x - 2) + 1\)
Simplifying the right side: \[ 3(x - 2) + 1 = 3x - 6 + 1 = 3x - 5 \] Therefore, the equation becomes: \[ 3x - 5 = 3x - 5 \] This is an identity, meaning it holds true for all values of \(x\). Thus, there are infinitely many solutions.
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Equation: \(3x - 5 = -9x + 15\)
Let's isolate \(x\): \[ 3x + 9x = 15 + 5 \] \[ 12x = 20 \] \[ x = \frac{20}{12} = \frac{5}{3} \] This has a single solution. Thus, there is one solution.
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Equation: \(3x - 5 = 3x + 10\)
Subtract \(3x\) from both sides: \[ 3x - 3x - 5 = 10 \] \[ -5 = 10 \] This is a false statement (since -5 does not equal 10), indicating there are no solutions. Thus, there are no solutions.
In summary:
- \(3x - 5 = 3(x - 2) + 1\) → Infinitely many solutions
- \(3x - 5 = -9x + 15\) → One solution
- \(3x - 5 = 3x + 10\) → No solutions