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Combining like terms is a fundamental technique in algebra that makes expressions easier to work with. When we say 'like terms', we are referring to terms that contain the same variable raised to the same power. For example, in the expression \(0.06x + 0.09(x + 5000)\), the terms \(0.06x\) and \(0.09x\) are considered like terms because they both contain the variable \(x\).
By combining these like terms, we simplify our algebraic expression. In this case, \(0.06x + 0.09x\) simplifies to \(0.15x\).
Here’s the step-by-step process to combine like terms:

• Identify the like terms in the expression. They should have the same variable and exponent.
• Add or subtract their coefficients (the numerical part attached to the variable). In our example, \(0.06 + 0.09 = 0.15\).
• Rewrite the expression with the combined term.

This process helps us to condense the expression and make further calculations simpler.

Distribution is an essential algebraic technique used to eliminate parentheses and spread multiplication across terms inside the parentheses. In the exercise, we started with \(0.09(x + 5000)\).
Here's what we did step by step:

• Multiply 0.09 by each term inside the parentheses separately. This gives us: \(0.09 \times x = 0.09x\).
• Next, \(0.09 \times 5000 = 450\).

After distributing the 0.09, we have: \(0.09x + 450\).
The distributive property is formally written as \(a(b + c) = ab + ac\). It ensures that all terms within the parentheses are multiplied by the term outside.

In algebra, coefficients are the numerical factors attached to variables. For example, in \(0.06x + 0.09(x + 5000)\), both 0.06 and 0.09 are coefficients.
Coefficients play a crucial role in combining like terms and distributing expressions. To combine like terms, you add or subtract their coefficients. To distribute a term across an expression, you multiply the coefficients by each term inside the parentheses.
Understanding coefficients helps you avoid mistakes in calculations and makes it easier to simplify expressions. In this case, combining \(0.06x\) and \(0.09x\) by adding their coefficients gives us \(0.15x\), and distributing \(0.09\) correctly helps to break down complex expressions into manageable parts.
When dealing with coefficients, always keep track of the signs (positive or negative) and ensure you perform arithmetic operations correctly.