Problem 69 - Intermediate Algebra

Question

Solve each equation. $$ 9 x=6 x $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0\).

Step 1: Understand the Equation

The equation given is \(9x = 6x\). We need to find the value of \(x\) that satisfies this equality.

Step 2: Subtract 6x from Both Sides

To isolate \(x\), start by subtracting \(6x\) from both sides of the equation. This simplifies to \(9x - 6x = 0\).

Step 3: Simplify the Equation

Now, simplify the expression by combining like terms on the left side: \(9x - 6x = 3x\). The equation becomes \(3x = 0\).

Step 4: Solve for x

To solve for \(x\), divide both sides of the equation by 3, giving \(x = 0/3\).

Step 5: Final Solution

Simplifying the division gives \(x = 0\). Thus, the solution to the equation is \(x = 0\).

Key Concepts

Isolating Variables

When solving algebraic equations, our goal is to find the value of the variable that makes the equation true. This process often involves isolating the variable on one side of the equation.

• Imagine you have an equation with terms containing the variable on both sides. Your first step is to bring all those terms to one side of the equation. This helps us to narrow down the equation to a simpler form, making it easier to solve.
• For instance, in the equation \(9x = 6x\), you need to get all \(x\) terms on one side. By subtracting \(6x\) from both sides, the equation transforms to \(9x - 6x = 0\).
• Remember, isolating the variable is like clearing the field to see exactly what value the variable holds. This is crucial for simplifying and eventually solving the equation.

Combining Like Terms

Combining like terms is a fundamental skill in solving algebraic equations. This involves simplifying the expression by adding or subtracting terms that have the same variable raised to the same power.

• Consider the expression \(9x - 6x\). Both terms are like terms because they contain the variable \(x\) to the first power.
• To combine them, you simply perform the basic arithmetic operation: \(9 - 6\), which leaves you with \(3x\). This reduces the equation to \(3x = 0\).
• By combining like terms, you not only simplify the expression but also move a step closer to finding the solution. This makes your equation much easier to work with.

Dividing Both Sides of an Equation

Once you've isolated the variable and combined like terms, you typically are left with a term sitting alongside the variable. The next step is to make the variable truly isolated and find its value.

• In our example, we have \(3x = 0\). Here, \(3x\) simply means \(3 \cdot x\). To isolate \(x\), you divide both sides of the equation by 3.
• Performing this division gives you \(x = \frac{0}{3}\), which simplifies to \(x = 0\). Dividing both sides helps in finding the exact value of the variable.
• Remember, whatever mathematical operation you perform on one side of an equation, you must also perform on the other side to maintain balance. This principle ensures that the equality holds throughout the solution process.