Problem 51 - Elementary Algebra

Question

The sum of \(9 x\) and 6 is 51 .

Step-by-Step Solution

Answer

The value of \(x\) is 5.

Step 1: Write the Equation

The problem states that the sum of \(9x\) and 6 is 51. This can be written as an equation: \(9x + 6 = 51\).

Step 2: Isolate the Term with Variable

Subtract 6 from both sides of the equation to isolate the term with the variable on one side. This gives: \(9x + 6 - 6 = 51 - 6\), simplifying to \(9x = 45\).

Step 3: Solve for the Variable

Divide both sides of the equation by 9 to solve for \(x\). So, \(x = \frac{45}{9}\).

Step 4: Simplify the Solution

Simplify \(x = \frac{45}{9}\) to find \(x = 5\).

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra, where we aim to find the value of the unknown variable that makes the equation true. An equation, like a balanced scale, must remain equal on both sides. To solve the equation \(9x + 6 = 51\), we must carefully and systematically work through a series of steps to determine the value of \(x\).

The overall objective is to isolate the variable, in this case \(x\), on one side of the equation.

• First, we structure the equation by arranging all of the known values and terms with variables on the correct sides.
• Then, through algebraic manipulation, we perform operations, such as adding, subtracting, multiplying, or dividing, to both sides equally.
• This ensures that the original balance of the equation is maintained while uncovering the value of the unknown.

By following these steps, we successfully solved for \(x = 5\) in the equation, validating our solution to the problem.

Simplifying Expressions

Simplifying expressions involves reducing complex mathematical statements to simpler, more manageable forms without changing their value. In the equation \(9x + 6 = 51\), simplifying plays a key role in solving for \(x\).

To accomplish this, we:

• Made the equation easier to work with by subtracting 6 from both sides, leading to \(9x = 45\).
• At each step, we simplified the expressions to bring us closer to isolating the variable.
• By reducing the equation to its simplest terms, it frequently becomes more apparent what operations should be performed next.

This process of simplification allows us to focus more clearly on solving for the variable, ultimately finding that \(x = 5\).

Algebraic Manipulation

Algebraic manipulation is the process of using algebraic techniques to transform and solve equations. It involves using operations such as addition, subtraction, multiplication, and division to rewrite and solve equations.

In our exercise, algebraic manipulation helped us to systematically work through each step. For instance:

• We subtracted 6 from both sides of the equation \(9x + 6 = 51\), simplifying it to \(9x = 45\).
• Then, by dividing both sides by 9, we isolated the variable \(x\), leading to \(x = 5\).

These techniques are powerful tools in algebra, allowing us to solve even the most complex equations. Mastering them not only helps in simplifying equations but also boosts overall mathematical problem-solving skills.