To solve the equation \(32(4 - 8x) + 5x = 1 - 7x\), we'll go through it step by step and justify each step using the properties of equality.
Step 1: Distribute on the left side
Starting with: \[ 32(4 - 8x) + 5x = 1 - 7x \]
We can distribute \(32\) on the left side: \[ 32 \cdot 4 - 32 \cdot 8x + 5x = 1 - 7x \]
Calculating the products: \[ 128 - 256x + 5x = 1 - 7x \]
Step 2: Combine like terms
Now, combine the like terms on the left hand side: \[ 128 - 251x = 1 - 7x \]
Step 3: Isolate the variable
Next, we want to isolate \(x\). We can start by moving all \(x\) terms to one side and constant terms to the other side. Let's add \(7x\) to both sides: \[ 128 - 251x + 7x = 1 \] This simplifies to: \[ 128 - 244x = 1 \]
Now, subtract \(128\) from both sides: \[ -244x = 1 - 128 \] This simplifies to: \[ -244x = -127 \]
Step 4: Solve for \(x\)
Now, divide both sides by \(-244\) to solve for \(x\): \[ x = \frac{-127}{-244} = \frac{127}{244} \]
Conclusion: Determine the number of solutions
The equation \(32(4 - 8x) + 5x = 1 - 7x\) simplifies to a single linear equation with one solution: \[ x = \frac{127}{244} \]
Thus, the number of solutions to the equation is one.
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