Teach: Expressions with Letters ③ Addition and Subtraction of Linear Expressions (Like Terms)

Expressions with Letters ③ Addition and Subtraction of Linear Expressions (Like Terms)

We will learn how to add and subtract expressions containing letters. The keyword is like terms. Learn the rule that terms with the same letters can be combined.

Terms, Coefficients, and Linear Expressions

Each part of an expression connected by addition is called a term.
　Example: The terms of 3x − 2y + 5 are 3x, −2y, and 5.

In a term containing a letter, the number multiplied by the letter is called the coefficient, and a term that does not contain a letter is called a constant term.
　Example: The coefficient of 3x is 3, the coefficient of −2y is −2, and 5 is a constant term.

A term containing only one letter raised to the first power is called a linear term, and an expression consisting only of such terms and constant terms is called a linear expression.
　Example: 3x − 7 and 4a + 1 are linear expressions. x² and ab are not linear expressions.

Example ①
For the expression −4x + 6 − 2y, state the terms, the coefficient of each term, and the constant term.
Answer: There are three terms: −4x, 6, and −2y.
The coefficient of −4x is −4, the coefficient of −2y is −2, and 6 is the constant term.

Like Terms and How to Combine Them

Like terms are terms whose letter parts are exactly the same. Like terms can be combined by adding or subtracting their coefficients.

  • 3x + 5x ＝ (3+5)x ＝ 8x
  • 4a − 7a ＝ (4−7)a ＝ −3a
  • 2x + 3y cannot be combined because x and y are different letters

When adding or subtracting linear expressions, combine like terms and constant terms separately.

Example ②
Calculate the following.
(1) 4x + 7 − 6x − 3　(2) (5a + 2) + (3a − 8)　(3) (4y − 5) − (y − 9)
Answer:
(1) x terms: 4x − 6x ＝ −2x, constants: 7 − 3 ＝ 4 → −2x + 4
(2) Remove the parentheses: 5a + 2 + 3a − 8. a terms: 5a + 3a ＝ 8a, constants: 2 − 8 ＝ −6 → 8a − 6
(3) Removing the parentheses from −(y − 9) gives −y + 9. Expression: 4y − 5 − y + 9. y terms: 4y − y ＝ 3y, constants: −5 + 9 ＝ 4 → 3y + 4

Note: When the expression before the parentheses is − (minus), removing the parentheses reverses the signs of all terms inside.
　Example: −(a − b + 3) ＝ −a + b − 3.

Key Points
  • Term: each part connected by addition / coefficient: the number multiplied by a letter / constant term: a term without a letter
  • Like terms have the same letter part. Combine them by adding or subtracting their coefficients
  • Different letters, such as x and y, cannot be combined
  • When the expression before the parentheses is “−”, removing them reverses the signs of all terms inside

Practice Problems

  1. Calculate the following: 6x − 3 − 2x + 8
  2. Calculate the following: (3a + 5) − (7a − 4)
  3. State the terms, the coefficient of x, and the constant term in the following expression: −5 + 3x − 2 − 4x

Answers and Explanations

  1. Answer: 4x + 5
    Explanation: x terms: 6x − 2x = 4x. Constants: −3 + 8 = 5. Therefore, 4x + 5.
  2. Answer: −4a + 9
    Explanation: Removing −(7a−4) gives −7a + 4. Expression: 3a + 5 − 7a + 4. a terms: 3a − 7a = −4a. Constants: 5 + 4 = 9. Therefore, −4a + 9.
  3. Answer: Combining like terms gives −x − 7. The terms are −x and −7. Coefficient of x: −1, constant term: −7
    Explanation: x terms: 3x − 4x = −x (coefficient: −1). Constants: −5 − 2 = −7. Although coefficient 1 is omitted, write −1 when stating the coefficient.

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