To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \), we will follow a systematic approach using properties of equality. Let's go step by step:
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Distribute on the left side:
\[ 32(4 - 8x) = 32 \cdot 4 - 32 \cdot 8x \]
Calculating each term:
\[ 32 \cdot 4 = 128 \] \[ 32 \cdot 8x = 256x \]
Therefore, the equation becomes:
\[ 128 - 256x + 5x = 1 - 7x \]
We can combine the \(x\) terms:
\[ 128 - 251x = 1 - 7x \]
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Move all terms involving \(x\) to one side and constant terms to the other side:
Let's add \(251x\) to both sides:
\[ 128 = 1 - 7x + 251x \]
Combine like terms on the right side:
\[ 128 = 1 + 244x \]
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Isolate the variable \(x\):
Subtract \(1\) from both sides:
\[ 128 - 1 = 244x \]
\[ 127 = 244x \]
Now, divide both sides by \(244\):
\[ x = \frac{127}{244} \]
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Check for uniqueness:
Since we found one specific value for \(x\) (i.e., \(x = \frac{127}{244}\)), we conclude that this equation has exactly one solution.
Summary:
- We used the distributive property to expand \(32(4 - 8x)\).
- We combined like terms using the properties of equality.
- We isolated \(x\) using addition and multiplication properties of equality.
- The conclusion is that the equation \(32(4 - 8x) + 5x = 1 - 7x\) has exactly one solution: \(x = \frac{127}{244}\).
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