Problem 56 - Elementary Algebra

Question

Simplify. $$ 15 y 2-34+710 y 2-12 $$

Step-by-Step Solution

Answer

The simplified expression is \(725y^2 - 46\).

Step 1: Group Like Terms

Identify the terms that can be grouped together. In this expression, there are terms involving \(y^2\) and constant numbers. The terms \(15y^2\) and \(710y^2\) have the same variable part \(y^2\), and the constants are \(-34\) and \(-12\).

Step 2: Combine Like Terms

Combine the terms involving \(y^2\) by adding them: \(15y^2 + 710y^2 = 725y^2\). Then, combine the constant terms: \(-34 - 12 = -46\).

Step 3: Write the Simplified Expression

After combining like terms, the expression simplifies to \(725y^2 - 46\).

Key Concepts

Like Terms

To understand algebraic expressions, it's crucial to grasp the concept of "like terms." In an expression, like terms are those that have the same variable parts raised to the same power. This means their variables and exponents must match exactly. For example:

• The terms \(5x^2\) and \(-3x^2\) are like terms since they both have \(x^2\).
• The terms \(2y\) and \(-7y\) are like terms as they both have just \(y\).
• However, \(x\) and \(x^2\) are not like terms because their exponents are different.

Identifying like terms is the first critical step in simplifying algebraic expressions because only like terms can be combined. Remember, constants, which are numbers without variables, are also like terms with other constants.

Combining Like Terms

Once you have identified the like terms in your expression, the next step is to "combine" them. Combining like terms means you add or subtract their coefficients while keeping the variable part intact. Consider the expression: \(15y^2 + 710y^2\).

• "Combine" here means to add the coefficients: \(15 + 710 = 725\).
• Since both terms have \(y^2\), the result is \(725y^2\).

Similarly, for constant terms like \(-34\) and \(-12\), simply add them as usual: \(-34 - 12 = -46\).
Combining like terms reduces the expression to fewer terms, making it easier to work with and understand.

Algebraic Simplification

Algebraic simplification involves both recognizing and combining like terms to create the most streamlined version of an expression. Simplified expressions are those where no further combining or factoring is possible, providing the simplest form of the expression.
For instance, in the example \(15y^2 - 34 + 710y^2 - 12\):

• We first combined the \(y^2\) terms: \(15y^2 + 710y^2 = 725y^2\).
• Then, we combined the constants: \(-34 - 12 = -46\).

These steps result in the simplified expression \(725y^2 - 46\). This is important because simpler expressions are easier to interpret, solve, and use in further calculations.
Always aim for the simplest form to enhance clarity and efficiency in problem-solving.