Question

Simplify \(\frac{5 x}{25 x+10 y}\)

Short answer

Answer: The simplest form of the expression is \(\frac{x}{5x+2y}\).

Step-by-step solution

Factor the numerator and the denominator

Factor out the common factors in the numerator and the denominator. In this case, we can factor out 5 from both the numerator and the denominator. Numerator: \(5x = 5\cdot x\). Denominator: \(25x + 10y = 5\cdot (5x+2y)\). So, the expression becomes: \(\frac{5\cdot x}{5\cdot (5x+2y)}\).

Cancel out the common factors

Now, we will cancel out the common factor, which is 5, from the numerator and the denominator. The expression becomes: \(\frac{x}{5x+2y}\).

Write the simplified expression

The simplified expression is \(\frac{x}{5x+2y}\).

Key concepts

Simplification

Simplifying algebraic fractions makes complex expressions easier to work with. By finding and canceling common factors in the numerator and the denominator, we can reduce fractions to their simplest form.
To start, identify any common factors in both parts of the fraction. This involves looking for numbers or variables that divide both the numerator and the denominator.

• Remember, simplifying does not change the value of the fraction; it only makes it simpler to perceive and manage.
• After identifying the common factors, you can "cancel" them, which means they can be removed from both the numerator and the denominator.

This results in a simplified expression, which is easier to understand and use in further calculations or transformations.

Factoring

Factoring is the process of breaking down an expression into a product of simpler terms or factors. This is a crucial step in simplifying algebraic fractions. It involves identifying the greatest common factor among the terms in the expression.
In the expression above, the factorization step took place as follows:

• The numerator, 5x, was factored into 5 times x.
• The denominator, 25x + 10y, was factored to reveal the greatest common factor, 5, giving us 5(5x + 2y).

Factoring is a foundational skill in algebra as it allows you to see the structure of expressions, making other algebraic manipulations, such as simplification and solving equations, more manageable.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. The key to working with rational expressions effectively is an understanding of simplification and factoring.
These types of expressions often require an extra step of algebraic manipulation to resolve.

• Firstly, ensure that both the numerator and denominator are factored completely.
• Only then can common factors be canceled to simplify the expression.

While simplifying rational expressions, it is important to check for restrictions. Restrictions occur where the denominator could potentially become zero, as dividing by zero is undefined in mathematics. Identifying these restrictions ensures the accuracy and correctness of the simplified expression.