Combining Like Terms

Mathematical Expressions and Simplification

Expression Analysis

• The original expression discussed in class is:

  • $3x + 4y - 5x - 92 - 17y$

Observations

• Number of Terms:

  • Total Number of Terms: Five

  • Total Number of Terms: Five (3x, 4y, -5x, -92, and -17y combine to give the final expression).

• Variables and Constants:

  • Types of variables observed:

    • Two variable types: x and y

    • One constant term identified as -92

  • Constant definition: A constant is a value that does not change.

  • Coefficient definition: A coefficient is a numerical factor multiplied by a variable in an algebraic expression. For example, in the expression 5x + 3y - 92, 5 is the coefficient of x, 3 is the coefficient of y, and -92 is the constant term that remains unchanged regardless of the values assigned to x and y.

• Definition of Like Terms:

  • Terms are considered like terms if they contain the same variable and exponent. For instance, $x^2$ and $x^2$ are like terms, whereas $x^2$ and $x^3$ are not.

  • Constant terms are also classified as like terms because their values can be combined (added or subtracted).

Combining Terms

• Original Expression: $3x + 4y - 5x - 92 - 17y$

• The goal of combining like terms is to simplify expressions by consolidating terms that have the same variable raised to the same power, thereby reducing the equation to its simplest form.

• Identifying Like Terms:

  • For terms involving the variable x: Circle 3x and -5x.

  • The sign before a term is crucial; it identifies whether it is positive or negative.

• Combining coefficients:

  • To combine like terms, the coefficients of the like terms will be added or subtracted following the rules of arithmetic:

    • Calculation:

      • Combine x terms:

      • $3x - 5x = -2x$

      • Or by viewing as: $3x + (-5x) = -2x$

      • Result: -2x

    • Combine y terms:

      • $4y - 17y = -13y$

      • Or as: $4y + (-17y) = -13y$

      • Result: -13y

    • Retaining the constant term: -92

• **Final Expression After Combination: **

  • The simplified expression after combining like terms will be:

  • $-2x - 13y - 92$

Important Concepts:

• The process of permutation in terms must follow the basic arithmetical rules of adding and subtracting coefficients while adhering to sign indications.

Additional Example and Explanation of Terms

• New Expression Introduced:

  • $-2x^2 - 2x + 6y - 7 + 6y^2 + 6x^2 - 7$

Observations

• Total Terms Count:

  • seven terms in the expression.

Combining Like Terms in Detail

• Identifying Like Terms:

  • Like Terms for $x^2$:

    • $-2x^2$ and $6x^2$ are like terms.

    • Combining yields:

    • Calculation:

      • $-2x^2 + 6x^2 = 4x^2$

  • Remaining Terms:

    • Identify remaining terms with no like counterparts:

      • For -2x, there are no like terms, thus it stays as is.

      • For y terms: 6y and y squared terms: 6y^2, the terms are not like because they have different exponents.

      • The constants -7 and -7 are like terms:

      • Combining constants:

        • $-7 + (-7) = -14$

Final Expression Order and Recommendations

• The rewritten expression taking into account the terms:

  • $4x^2 - 2x + 6y^2 - 14$

• Descending Order of Terms:

  • Can be recommended that when writing expressions, they may be presented in descending order of variable exponents to maintain clarity and consistency. Example of proper order:

  • $4x^2 + 6y^2 - 2x - 14$

  • Importance of maintaining consistency with signs.

  • Confirmed that both forms of the expression are valid, but descending order is preferred for higher-level clarity in mathematics.

Order of Terms and Significance

• Order Is Not Fixed:

  • Students noted it's acceptable to list terms in any order as long as appropriate signs (+/-) are appropriately used.

  • Maintaining the accuracy of the signs is essential in conveying the correct value of each term involved.