Problem 26 - Algebra 1: Concepts and Skills

Question

Simplify the expression by combining like terms if possible. If not possible, write already simplified. $$8+2 z+4+3 z$$

Step-by-Step Solution

Answer

The simplified form of the equation \(8+2 z+4+3 z\) is \(12 + 5z\).

Step 1: Identify Like Terms

Observe the given equation \(8+2z+4+3z\). Identify the like terms. Here the terms \(8\) and \(4\) are like terms because they are constants meaning they do not contain a variable. On the other hand, the terms \(2z\) and \(3z\) are like terms because they both contain the variable 'z'.

Step 2: Combine Like Terms

Combine the like terms separately. Start with the constants: \(8 + 4 = 12\). Next, you combine the terms with 'z': \(2z + 3z = 5z\).

Step 3: Write Simplified Expression

After combining like terms, bring together the simplified terms to form the simplified equation which is \(12 + 5z\)

Key Concepts

Simplify Expression

When faced with an algebraic expression, such as \( 8+2z+4+3z \) , the goal is to make it as straightforward and compact as possible. This process is known as simplifying the expression. Simplification helps in understanding and working with expressions more effectively by reducing complexity and creating a more direct equation or expression.

To simplify an expression, one must follow these general steps:

• Identify like terms.
• Combine those like terms.
• Rewrite the expression in its simplest form.

In the given exercise, \( 8+2z+4+3z \) was made simpler by adding the constants, giving us \( 12 \) , and by combining the coefficients of \( z \) , which resulted in \( 5z \) . This process led to a simplified expression of \( 12+5z \) , which is much easier to work with in algebraic operations.

Like Terms Algebra

In algebra, like terms are terms that contain the same variables, raised to the same power. The concept is crucial when we are organizing and simplifying algebraic expressions. In the example problem, the given expression can be broken down into terms involving constants, \( 8 \) and \( 4 \) , and terms involving the variable \( z \) , namely \( 2z \) and \( 3z \) .

Only like terms can be combined; this is why \( 8 \) and \( 4 \) are added together, as well as \( 2z \) and \( 3z \) . When combining like terms, you are essentially performing basic arithmetic operations to merge the coefficients of the terms. This helps in streamlining equations and sets the stage for further algebraic manipulation.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols that represent mathematical relationships. These expressions can range from simple, such as \( 2 + 3 \) , to complex, involving multiple variables and powers, like \( 4x^2 + 3xy - 5y + 7 \) .

An understanding of algebraic expressions is foundational for students as they advance in their study of algebra. The expression provided in the task, \( 8+2z+4+3z \) , is a linear algebraic expression because it does not contain any variables raised to a power higher than one and it is in one variable, \( z \) .

Recognizing different types of terms—constants, coefficients, and variables—is essential for the process of simplifying expressions. Whether an expression is already simplified or can be further simplified depends on whether it has like terms that can be combined. If no like terms exist, the expression is considered as already being in its simplest form.