Problem 69 - Algebra 1

Question

Solve the equation if possible. $$ 9-5 z=-8 z $$

Step-by-Step Solution

Answer

The solution to the equation is \( z = 3 \).

Step 1: Rearranging the equation

Start by moving all terms containing \( z \) to one side of the equation and remaining terms to the other side. This can be done by adding \( 5z \) to both sides of the equation. The equation then becomes: \( 9 = 3z \).

Step 2: Solving for \( z \)

Finally, to solve for \( z \), divide both sides of the equation by 3. This gives: \( z = \frac{9}{3} \).

Key Concepts

Solving Equations

Solving a linear equation involves finding the value of the variable that makes the equation true. Linear equations typically involve terms with variables raised to the first power, constants, and operations like addition, subtraction, multiplication, or division.

To tackle linear equations:

• Look for like terms, which are terms that contain the same variable raised to the same power.
• Move all terms involving the variable to one side of the equation by using addition or subtraction.
• Perform algebraic operations to isolate the variable on one side, solving for its value.

A step-by-step process involves manipulating the equation logically, so at each stage, the equation remains balanced until the variable is isolated on one side.

Variables

Variables are symbols, often represented by letters like \( z \), that stand for unknown values in mathematical expressions and equations. They are essential tools in algebra because they allow for generalization and abstraction. In the equation \( 9 - 5z = -8z \), the variable \( z \) represents the unknown number we aim to find.

Understanding variables involves:

• Identifying them within the equation.
• Realizing that they can take on various values, which we determine through solving.
• Appreciating their role as placeholders that can simplify complex real-world problems into manageable algebraic expressions.

Variables allow expressions to be transformed and solved, enabling us to find specific solutions or even a range of solutions.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations or expressions to find unknown values. This involves techniques such as:

• Adding or subtracting the same number or term from both sides of an equation to maintain balance.
• Combining like terms, which makes it simpler to isolate the variable.
• Using multiplication or division to solve for the variable when it is a coefficient in the equation.

For instance, in the given equation \( 9 - 5z = -8z \), algebraic manipulation allows us to move \( 5z \) to the other side, turning it into \( 9 = 3z \).

Such manipulations are crucial because they provide the pathway to simplify complex problems systematically, ensuring that the relationships between variables and constants are preserved.