Problem 4 - Algebra 1

Question

Solve the equation. $$6 x=18$$

Step-by-Step Solution

Answer

The solution to the equation \(6x = 18\) is \(x = 3\).

Step 1: Identify the Equation

The provided problem is an equation, which is \(6x = 18\) . We are tasked to find the value of the variable \(x\) that would satisfy this equation.

Step 2: Divide Both Sides by 6

The variable \(x\) is currently being multiplied by 6. To isolate \(x\), divide both sides of the equation by 6. Doing this gives us the new equation \(x = 18/6\).

Step 3: Solve for x

To find the value of \(x\), we must perform the division on the right side of the equation. \(18 \div 6 = 3\), therefore, \(x = 3\) is the solution of the equation.

Key Concepts

Isolate the Variable

One of the foundational steps in solving linear equations is to isolate the variable. To do this, we aim to get the variable (in this case, \( x \)) alone on one side of the equation. This allows us to easily identify its value.

To isolate \( x \), observe the equation \( 6x = 18 \). The variable \( x \) is currently accompanied by the number 6 through multiplication. Our goal is to eliminate the 6 from the left side.

By reflecting on the inverse operations in algebra, realize that the opposite of multiplication is division. Therefore, dividing both sides of the equation by 6 is the perfect move. This operation will effectively cancel out the 6, thus freeing \( x \) on one side of the equation. Isn’t it fascinating how isolating a variable works like a puzzle pairing matching pieces to uncover the missing number?

Division in Algebra

Division plays a crucial role in algebra, especially when working to isolate variables.

In the equation \( 6x = 18 \), \( x \) is bonded with 6 by multiplication. To find \( x \), we must divide through this bond using division. This is akin to undoing what was done to the variable, returning it to its simplest form.

Performing the division \( \frac{18}{6} \) simplifies to 3. The division process:

• Determine the operation needed: Since the variable is multiplied, divide to reverse the operation.
• Calculate accurately: Double-check the division calculation to ensure the result is correct.

This solution enables the simplification and intuitiveness in manipulating algebraic expressions safely.

Basic Algebraic Operations

Algebra involves several fundamental operations, including addition, subtraction, multiplication, and division.

These operations allow us to transform and solve equations. In our equation, \( 6x = 18 \), basic operations are keys to finding \( x \).

In most algebraic equations:

• Addition and subtraction: These usually involve moving numbers across the equals sign to combine or remove terms.
• Multiplication and division: Essential for scaling variables and breaking down coefficients.

Exercise these operations consistently and purposefully. Knowing when and how to apply each ensures effective problem-solving and boosts confidence in managing more complex algebraic expressions.