Problem 57 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 2 z+4 a b+5 z-a b+12(1-a b-z) $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is $$-5z -9ab + 12$$.

Step 1: Distribute the constant 12

To distribute the 12, multiply each term inside the parenthesis by 12: $$ 12(1-ab-z) = 12 - 12ab - 12z $$

Step 2: Combine the expression with the distributed terms

Now, we have to add the distributed terms to the original algebraic expression: $$ 2z + 4ab + 5z - ab + 12 - 12ab - 12z $$

Step 3: Combine like terms

Next, we will combine the like terms in the expression: $$ (2z + 5z - 12z) + (4ab - ab - 12ab) + 12 $$

Step 4: Simplify the expression

Finally, we just need to simplify the expression by performing the operations: $$ -5z - 9ab + 12 $$ The simplified algebraic expression is: $$ -5z -9ab + 12 $$

Key Concepts

Distributive Property

Understanding the distributive property is important when simplifying algebraic expressions. This property states that multiplying a number by a group of numbers inside parentheses is the same as doing each multiplication separately and then adding the results.
For example, consider this exercise where we distribute the constant 12 across the expression inside the parentheses:

• Start with: \(12(1-ab-z)\)
• Multiply 12 by each term: \(12 \times 1, \; 12 \times (-ab), \; 12 \times (-z)\)
• The result is: \(12 - 12ab - 12z\)

This helps break down more complicated expressions into larger, easier-to-manage pieces, setting the stage for further simplification.

Combining Like Terms

Combining like terms means merging terms in an expression that have the same variables raised to the same power, simplifying the expression more.

• Like terms have identical variable parts. For instance, \(2z\), \(5z\), and \(-12z\) are like terms because they all contain the variable \(z\).
• Similarly, \(4ab\), \(-ab\), and \(-12ab\) are also like terms because they all involve the same product of variables \(ab\).

In the step-by-step solution for this problem:

• We combine \(2z + 5z - 12z\) into \(-5z\) because they all involve the variable \(z\).
• Similarly, \(4ab - ab - 12ab\) simplifies to \(-9ab\).

Combining like terms helps greatly in making the algebraic expression simpler and easier to interpret.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and operators. They represent mathematical phrases and are foundational in algebra.
An algebraic expression, like \(2z + 4ab + 5z - ab + 12(1-ab-z)\), consists of:

• Constants: numbers without variables, like 12.
• Variables: letters representing unknown values, such as \(z\) and \(ab\).
• Operators: include addition \(+\), subtraction \(-\), multiplication (implied by placing terms next to each other or using parentheses), and division.

The goal is often to simplify these expressions, making them more straightforward for solving specific problems or equations. Understanding how constants, variables, and operators interact is key to mastering algebraic simplification.