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Simplifying expressions is all about making them easier to work with by combining like terms and reducing complexity. To start with, look for terms that are similar, which are generally defined as "like terms." Then, perform arithmetic operations such as addition or subtraction on their coefficients.
The process of simplifying typically involves:

• Identifying like terms within the expression.
• Grouping and combining these like terms.
• Eliminating unnecessary parts of the expression to provide clarity.

By simplifying expressions, you make it easier to understand and solve algebraic problems, allowing for quicker calculations and a better grasp of the mathematical relationships within the expression.

Like terms are essentially the building blocks of an algebraic expression that can easily be combined. They are terms that have the same variables raised to the same power. For instance, in the expression from the exercise, the terms \(7x\) and \(-9x\) are like terms because they both contain the variable 'x' with the same exponent.
Similarly, the terms \(-5y\) and \(19y\) are like terms because they share the variable 'y'.

• The exponents of the variables must match for terms to be considered like terms.
• Only the coefficients, the numerical part of the terms, differ and are subject to arithmetic operations.

Combining like terms simplifies the expression, making it more streamlined and sometimes transforming it into an easily solvable equation.

Coefficients are the numerical part of the terms in algebraic expressions. They multiply the variable part of the term and play a crucial role in operations involving like terms. For example, in the term \(7x\), 7 is the coefficient. Understanding coefficients can help with simplifying expressions, as you merely add or subtract these numbers when combining like terms.
To simplify using coefficients as shown in the exercise:

• Add or subtract the coefficients of like terms only.
• Be mindful of the sign (positive or negative) in front of each coefficient, as this affects the summation.

For instance, in the combination \(7x + (-9x)\), the coefficients 7 and -9 are combined to give -2, resulting in \(-2x\) for that part of the expression.
Understanding coefficients is essential for mastering algebraic manipulations and simplifying expressions accurately.