Problem 4 - Elementary Algebra

Question

Simplify each of the following expressions. $$ 5 a+2 b+4 a-b-7 b $$

Step-by-Step Solution

Answer

Question: Simplify the following algebraic expression: $$5a + 2b - b + 4a - 7b$$ Answer: $$9a - 6b$$

Step 1: (Step 1: Identify like terms)

(In the given expression, \(5a\) and \(4a\) are like terms because they have the same variable 'a'. Similarly, \(2b\), \(-b\), and \(-7b\) are like terms because they have the same variable 'b'.)

Step 2: (Step 2: Combine the like terms)

(Now, we can add or subtract the coefficients of the like terms. For the terms with \(a\), we have \(5a + 4a\). For the terms with \(b\), we have \(2b - b - 7b\). )

Step 3: (Step 3: Calculate the sums)

(Add the coefficients of the like terms with \(a\): \(5a + 4a = 5 + 4 \cdot a = 9a\). Subtract the coefficients of the like terms with \(b\): \(2b - b - 7b = 2 - 1 - 7 \cdot b = -6b\). )

Step 4: (Step 4: Write the simplified expression)

(Now we can write the simplified expression as a combination of the simplified terms with \(a\) and \(b\): $$9a - 6b$$. ) Therefore, the simplified expression is $$ 9 a-6 b $$.

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, a fundamental skill is combining like terms. Like terms are terms that have exactly the same variables raised to the same powers, even if the coefficients (the numbers multiplying the variables) are different. In the exercise, 5a and 4a are like terms, as are 2b, -b, and -7b. To combine them, we add or subtract their coefficients.

If you’re working on an algebra problem and trying to combine like terms, here's a helpful tip: underline or circle the like terms, which can help you visually group them for easier calculation. After identifying the like terms in our example, we add the coefficients for a terms (5 and 4) to get 9a and for b terms (2, -1, and -7) to get -6b. Remember, the minus sign before a term indicates the coefficient is negative.

Algebraic Coefficients

In algebra, the numbers that multiply the variables are called algebraic coefficients. These coefficients play a crucial role when simplifying expressions. They determine how much a term will contribute to the final result. In the exercise, the terms had coefficients of 5, 4, 2, -1, and -7.

When you encounter a problem, focus on correctly identifying and handling these coefficients, especially when negative numbers are involved. For example, the term -b can be thought of as -1 * b, because any number without a visible coefficient has an implied coefficient of 1. Understanding this will allow you to combine terms accurately to reach the simplified expression 9a - 6b.

Elementary Algebra

At the heart of mathematics is elementary algebra, which involves operations on algebraic expressions, including simplification, factoring, and solving equations. Simplifying expressions, like we did in the exercise, is a fundamental skill that can make math much less daunting. It's all about applying basic arithmetic operations—addition, subtraction, multiplication, and division—to symbols instead of numbers.

To improve in elementary algebra, practice the distributive property and combining like terms regularly. Remember to double-check your work for mistakes, such as combining unlike terms or incorrectly adding coefficients. As you become more comfortable with these concepts, more complex areas of algebra will become much more approachable.