To solve the equation \(32(4 - 8x) + 5x = 1 - 7x\) and justify the steps using properties of equality, follow these steps:
Step 1: Distribute
Distribute the \(32\) across \((4 - 8x)\):
\[ 32 \cdot 4 - 32 \cdot 8x + 5x = 1 - 7x \]
Calculating the terms:
\[ 128 - 256x + 5x = 1 - 7x \]
Step 2: Combine like terms
Combine the \(x\) terms on the left side:
\[ 128 - 251x = 1 - 7x \]
Step 3: Get all \(x\) terms on one side
Add \(7x\) to both sides:
\[ 128 - 251x + 7x = 1 \]
This simplifies to:
\[ 128 - 244x = 1 \]
Step 4: Isolate the constant
Subtract \(128\) from both sides:
\[ -244x = 1 - 128 \]
Calculating the right side:
\[ -244x = -127 \]
Step 5: Solve for \(x\)
Divide both sides by \(-244\):
\[ x = \frac{-127}{-244} \]
This simplifies to:
\[ x = \frac{127}{244} \]
Conclusion
The solution to the equation is \(x = \frac{127}{244}\). Since we found a single unique solution, the number of solutions to the equation is 1.
Justification with Properties of Equality
- Distributive Property: Used to distribute \(32\) across the expression \( (4 - 8x) \).
- Combining Like Terms: Followed to simplify both sides of the equation.
- Addition Property of Equality: Applied to add \(7x\) to both sides and \(128\) to isolate the variable.
- Division Property of Equality: Used to divide both sides by \(-244\) to solve for \(x\).
Thus, we have a logical progression of steps that demonstrate the equality and justify the solution methodically.
1 answer