To determine whether the equation \(-3(8x + 10) = -14x + 2\) has exactly one solution, let's solve it step-by-step.
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Distribute on the left side: \[ -3(8x + 10) = -24x - 30 \] So the equation becomes: \[ -24x - 30 = -14x + 2 \]
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Rearrange the equation to isolate the variable \(x\): First, add \(24x\) to both sides: \[ -30 = 10x + 2 \] Now, subtract \(2\) from both sides: \[ -30 - 2 = 10x \] That simplifies to: \[ -32 = 10x \]
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Solve for \(x\): Divide both sides by \(10\): \[ x = -\frac{32}{10} = -\frac{16}{5} \]
This process shows that the equation has exactly one solution, namely: \[ x = -\frac{16}{5} \]
Summary of Conditions for Exactly One Solution:
An equation will have exactly one solution when it can be simplified to a form where the variable appears with a non-zero coefficient, ensuring that there's a unique \(x\) that satisfies the equation. In this case, we found one unique value of \(x\).
Final answer: \[ x = -\frac{16}{5} \]
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