Problem 30 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 10 y-15 y $$

Step-by-Step Solution

Answer

Answer: The simplified form of the given algebraic expression is "- 5y".

Step 1: Identify the like terms

In this expression, the like terms are "10y" and "-15y". Both terms have the same variable, which is y, and their coefficients are 10 and -15, respectively.

Step 2: Combine the like terms

To combine the like terms, we need to subtract the coefficients 10 and -15. This can be written as: $$ 10y - 15y $$

Step 3: Subtract the coefficients

Now, we subtract the coefficients 10 and 15: $$ 10 - 15 = -5 $$

Step 4: Write the simplified expression

As a result, the simplified expression will be -5 times the variable y: $$ - 5y $$ So, the simplified algebraic expression is: $$ - 5y $$

Key Concepts

Like Terms

In algebra, recognizing and working with like terms is essential for simplifying expressions. Like terms are terms that have the exact same variable raised to the same power. This means that only the coefficients, the numerical part in front of the variables, need to be considered during simplification.

A quick way to identify like terms is to look for terms that share the same letters and exponents. For example, in the expression "10y - 15y," both terms contain the variable "y". This makes them like terms. Since they share the same variable, we can proceed to combine them.

Remember this:

• The variable, including any exponents, must match perfectly for terms to be considered like terms.
• Only like terms can be combined through addition or subtraction.

By correctly identifying and combining like terms, you'll simplify algebraic expressions more effectively.

Coefficients

Coefficients are the numbers that multiply the variable within a term. They play a crucial role in combining like terms. In the expression "10y - 15y," we focus on the numbers 10 and -15, which are the coefficients of the variable "y".

To combine like terms, you manage their coefficients. Subtract or add them according to the algebraic expression. Thus, in "10y - 15y," you subtract 15 from 10, which gives you -5. This new coefficient is then multiplied by the shared variable to produce a simplified term.

Here's what to remember about coefficients:

• They dictate the number of times the variable is counted in a term.
• You only adjust coefficients when simplifying an expression; the variable itself remains unchanged until all similar terms are combined.

Adequately grasping how coefficients interact with variables is key to mastering algebraic expressions.

Variable Terms

Variable terms in algebra are parts of an expression that include variables. In the expression "10y - 15y", both terms "10y" and "-15y" are variable terms. These terms consist of two parts: the coefficient and the variable.

Difference in variable terms from other algebraic terms lies in their inclusion of variables (like y or x). These variables are the parts of an expression that can change or vary, which is why they are referred to as 'variable.'

Understanding variable terms in algebra boils down to:

• Combining terms with the same variable: Only terms with the same variable component can be combined.
• Seeing equations as collections of these terms that you can manipulate to find equivalent expressions.

Once you are comfortable with what makes up a variable term, regrouping and simplifying them becomes far easier.

Algebraic Simplification

Simplifying an algebraic expression involves reducing it to its simplest form, making it easier to work with. The goal is to combine like terms and perform arithmetic on their coefficients until you can't simplify any further.

In our example "10y - 15y," simplification is done by performing the operation between the coefficients of like terms. By subtracting 15 from 10, we get -5, resulting in the simplified expression "-5y."

Key points about algebraic simplification include:

• Always first identify and combine like terms.
• Handle the coefficients with the appropriate arithmetic operations.
• The variable stays the same through the process, only the coefficients change.

Algebraic simplification helps in making equations more manageable, especially when solving more complex problems. It is a fundamental skill for algebraic problem-solving.