Problem 84 - Algebra A Combined Function

Question

Solve each equation. $$ 3 x=0 $$

Step-by-Step Solution

Answer

\(x = 0\)

Step 1: Understand the Equation

The given equation is a simple linear equation: \(3x = 0\). This means that a single variable, \(x\), is being multiplied by 3 to result in 0.

Step 2: Isolate the Variable

To solve for \(x\), we need to isolate the variable on one side of the equation. We can do this by dividing both sides of the equation by 3. This step gives us: \(x = \frac{0}{3}\).

Step 3: Simplify the Equation

Divide the right side of the equation by 3. Since any number divided by 3, where the numerator is 0, results in 0, we have \(x = 0\). Thus, the solution to the equation is \(x = 0\).

Key Concepts

Solving Equations

When we talk about solving linear equations like \(3x = 0\), our goal is to find the value of \(x\) that makes the equation true. This process involves a few straightforward steps designed to uncover this unknown number.
Linear equations usually contain terms connected by an equals sign. The central idea is that both sides are balanced, meaning whatever you do to one side must be done to the other to maintain this balance.
To solve an equation:

• First, identify the equation type. Here, \(3x = 0\) is a linear equation because the variable \(x\) is not raised to any power other than one.
• Next, simplify the equation through transformation steps until the solution to the variable is clear.

This covers the fundamental approach to solving equations.

Isolation of Variables

Isolating variables is a vital skill in solving equations. The goal here is to get the variable \(x\) by itself on one side of the equation.

For \(3x = 0\), we need \(x\) by itself.
Here's how:

• Recognize that \(x\) is multiplied by 3. To cancel out this multiplication, we perform the inverse operation, which is division.
• Divide both sides by 3: \(x = \frac{0}{3}\).

This action leaves \(x\) isolated, showing that \(x\) equals whatever is on the other side of the equation post-division. Successfully isolating the variable is a crucial step toward solving equations accurately.

Arithmetic Simplification

Arithmetic simplification involves reducing expressions to their simplest form. This step is crucial to making the equation easier to interpret and solve.

In our example \(x = \frac{0}{3}\), we perform arithmetic simplification as follows:

• Since 0 divided by any non-zero number still equals 0, the equation simplifies to \(x = 0\).

This final simplification step tells us the solution in its simplest, clearest form. Efficient arithmetic operations play a key role in making complex equations much more manageable.