Let's solve the equation \(-9 + 6x = -3(3 - 2x)\) step by step, indicating the properties used at each step.
Step 1: Distribute on the right side of the equation. \[ -3(3 - 2x) = -3 \cdot 3 + (-3) \cdot (-2x) = -9 + 6x \]
- Property used: Distributive Property
Now the equation looks like this: \[ -9 + 6x = -9 + 6x \]
Step 2: Add \(9\) to both sides to eliminate the constant on the left. \[ -9 + 6x + 9 = -9 + 6x + 9 \] This simplifies to: \[ 6x = 6x \]
- Property used: Addition Property of Equality
Step 3: Subtract \(6x\) from both sides to isolate variables. \[ 6x - 6x = 6x - 6x \] This simplifies to: \[ 0 = 0 \]
- Property used: Subtraction Property of Equality
Step 4: Interpret the result. Since \(0 = 0\) is a true statement, this means the equation holds for all values of \(x\). Thus, we conclude that the equation has infinitely many solutions.
- Property used: Identity Property (since any number equals itself).
Final Summary:
- We used the Distributive Property, Addition Property of Equality, Subtraction Property of Equality, and the Identity Property to solve the equation. The final outcome indicates that the equation is always true regardless of \(x\).