Question

Simplify the expression. Assume all variables are positive. \(\frac{12 x}{4 x}+5 x\)

Short answer

The simplified expression is \(3 + 5x\).

Step-by-step solution

Simplify the fraction

The fraction in the expression is \(\frac{12 x}{4 x}\). Notice that both the numerator and the denominator are divisible by \(4x\). Dividing both by \(4x\) simplifies the fraction to 3.

Rewrite the original expression replacing the fraction

Substitute the simplified fraction back into the original expression. The original expression now becomes: 3 + \(5x\).

Final Simplified Expression

There are no like terms in the expression 3 + \(5x\), so that is the simplest form.

Key concepts

Simplifying Fractions

Simplifying fractions is a fundamental concept in algebra that allows us to reduce expressions to their simplest form. This not only makes calculations easier to manage but also helps us to better understand the relationship between numbers. In the context of our problem, we see that the fraction \(\frac{12x}{4x}\) consists of a numerator (12x) and a denominator (4x) which both contain the variable 'x'. To simplify a fraction, we look for common factors in the numerator and the denominator.

In this case, 'x' is a common factor, and further, we can divide both terms by the number 4. When we divide 12 by 4, we get 3, and when we divide 'x' by itself, it simplifies to 1, because anything divided by itself equals 1 (except for zero). Therefore, \(\frac{12x}{4x}\) simplifies to just 3. Remember that when simplifying fractions, we should always look for the greatest common factor to make the simplification process as efficient as possible.

Algebraic Operations

Algebraic operations refer to the processes we use to manipulate algebraic expressions. These include addition, subtraction, multiplication, and division, much like arithmetic operations, but with the addition of variables and sometimes exponents. When improving our exercise solutions, focusing on algebraic operations includes not just performing the operation, but also understanding the 'why' behind each step.

For instance, in the original problem \(\frac{12x}{4x}+5x\), we use division to simplify the fraction and then addition to combine the constant and the variable term. The key to successfully carrying out algebraic operations is to perform them step by step, always applying rules such as the distributive law or the associative and commutative properties as needed. Furthermore, explaining the reasoning behind each step can help clarify any confusion and reinforce understanding.

Like Terms in Algebra

Like terms in algebra are terms that contain the same variable raised to the same power. Combining like terms is a crucial strategy used to simplify algebraic expressions and solve equations. However, it's important to note that constants (numbers without variables) and terms with different variables or exponents are not considered like terms and cannot be combined.

In our example, after simplifying the fraction, we obtained 3 + \(5x\). Here, 3 is a constant, and \(5x\) is a term with a variable. Since they are not like terms, they cannot be combined further. It's essential to understand this principle to avoid common mistakes in simplification. Recognizing like terms allows for more accurate and streamlined expressions, paving the way for easier problem-solving in algebra.