To determine which of the provided equations is equivalent to \(4x + 6 = 1\), we can start by rearranging the original equation to form a standard equation for comparison.
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Start with the original equation:
\[ 4x + 6 = 1 \]
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Subtract 6 from both sides:
\[ 4x = 1 - 6 \] \[ 4x = -5 \]
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Now, we can solve for \(x\) by dividing both sides by 4:
\[ x = -\frac{5}{4} \]
Next, we'll evaluate each option to see if they also yield \(x = -\frac{5}{4}\).
Option A: \(1 + 3(x + 2) = 1\)
Set the equation:
\[ 1 + 3(x + 2) = 1 \]
Subtract 1 from both sides:
\[ 3(x + 2) = 0 \]
Divide by 3:
\[ x + 2 = 0 \]
Subtract 2:
\[ x = -2 \] (This does not match \(-\frac{5}{4}\).)
Option B: \(5(x + 2) - x = 1\)
Set the equation:
\[ 5(x + 2) - x = 1 \]
Distribute the 5:
\[ 5x + 10 - x = 1 \]
Combine like terms:
\[ 4x + 10 = 1 \]
Subtract 10 from both sides:
\[ 4x = 1 - 10 \] \[ 4x = -9 \]
Divide by 4:
\[ x = -\frac{9}{4} \] (This does not match \(-\frac{5}{4}\).)
Option C: \(18 + 6(x - 2) - 2x = 1\)
Set the equation:
\[ 18 + 6(x - 2) - 2x = 1 \]
Distribute the 6:
\[ 18 + 6x - 12 - 2x = 1 \]
Combine like terms:
\[ (6x - 2x) + (18 - 12) = 1 \] \[ 4x + 6 = 1 \]
This is the same as the original equation! (This matches our rearranged equation.)
Option D: \(9x - 5(x - 1) - 1 = 1\)
Set the equation:
\[ 9x - 5(x - 1) - 1 = 1 \]
Distribute the -5:
\[ 9x - 5x + 5 - 1 = 1 \]
Combine like terms:
\[ (9x - 5x) + (5 - 1) = 1 \] \[ 4x + 4 = 1 \]
Subtract 4 from both sides:
\[ 4x = 1 - 4 \] \[ 4x = -3 \]
Divide by 4:
\[ x = -\frac{3}{4} \] (This does not match \(-\frac{5}{4}\).)
Conclusion: The only option that is equivalent to the original equation \(4x + 6 = 1\) is:
C. \(18 + 6(x - 2) - 2x = 1\).
1 answer