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Subtraction of algebraic expressions is a fundamental concept in algebra that involves the operation of subtracting one algebraic expression from another. This process is important for solving various mathematical problems and simplifying expressions. This topic includes key rules and properties such as the distributive property and the concept of combining like terms. Understanding of subtraction of algebraic expressions is not only important for competitive exams preparation, but also for higher-level mathematics and various real-world applications.Continue reading to learn more!
Table of Contents
What is Subtraction Algebraic Expression?
An algebraic expression consists of variables, constants, and operations (addition, subtraction, multiplication, and division). When subtracting algebraic expressions, you combine or simplify these expressions by subtracting the terms of one expression from those of another.
Components:
- Terms: Parts of the expression separated by plus or minus signs.
- Coefficients: Numerical factors of terms.
- Variables: Symbols representing numbers (like a, b, x, y, etc.).
- Constants: Fixed numerical values.
Method:
- Distribute Negative Sign: If needed, distribute the negative sign across the terms in the expression being subtracted.
- Combine Like Terms: Group and simplify terms with the same variable and exponent.
Rules:
- Distributive Property: For an expression like a−(b+c), you can rewrite it as a−b−c.
- Combining Like Terms: Terms with the same variable and exponent are combined.
Example:
- To subtract 2x+3 from 5x−4, write it as (5x−4)−(2x+3).
- Distribute the negative sign: 5x−4−2x−3.
- Combine like terms: (5x−2x)+(−4−3)=3x−7.
Components of Subtraction Algebraic Expressions
Before diving into subtraction, let’s understand the building blocks of algebraic expressions:
- Terms: These are the individual components of an algebraic expression separated by addition or subtraction signs. For example, in the expression 3x + 2y – 5, the terms are 3x, 2y, and -5.
- Coefficients: The numerical factor accompanying a variable in a term is called a coefficient. In 3x, the coefficient is 3.
- Variables: These are letters or symbols that represent unknown values. In 2y, the variable is y.
- Constants: These are numbers without variables, representing fixed values. In -5, the constant is -5.
- Like Terms: Terms that have the same variables raised to the same powers are called like terms. For example, 3x and -2x are like terms.
- Unlike Terms: Terms that have different variables or the same variables raised to different powers are unlike terms. For example, 3x and 2y are unlike terms.
Also Read: Rational And Irrational Numbers: Differences, Examples
Methods to Solve Subtraction of Algebraic Expressions
There are two primary methods to subtract algebraic expressions:
Horizontal Method
- Write the expressions: Arrange the given expressions in a horizontal format.
- Change signs: Change the sign of each term in the expression being subtracted.
- Combine like terms: Group and combine the like terms based on their signs.
- Simplify: Perform the necessary calculations to obtain the final result.
Column Method
- Arrange vertically: Write the expressions one below the other, aligning like terms in columns.
- Change signs: Change the signs of all terms in the expression being subtracted.
- Combine columns: Add the terms in each column.
- Write the result: Combine the results from each column to form the final answer.
Rules of Subtraction of Algebraic Expressions
When subtracting algebraic expressions, following specific rules helps ensure accuracy and consistency. Here are the key rules:
- Distribute the Negative Sign:
- If the expression being subtracted is within parentheses, distribute the negative sign across all terms inside the parentheses.
- Example: For (2x+3)−(4x−5), distribute the negative sign to get 2x+3−4x+5.
- Combine Like Terms:
- After distributing, combine terms that have the same variable and exponent. Like terms are terms with identical variables raised to the same power.
- Example: From 2x+3−4x+5, combine 2x−4x and 3+5 to get −2x+8.
- Perform Arithmetic Operations:
- Carry out addition or subtraction of constants and coefficients of like terms as needed.
- Example: In −2x+8, the final expression is simplified with no further like terms to combine.
- Keep the Order of Operations:
- Follow the order of operations (PEMDAS/BODMAS) as you distribute and combine terms. This ensures that subtraction is handled correctly in the context of the expression.
- Treat Subtraction as Adding the Opposite:
- Subtracting a term is equivalent to adding its negative. This can simplify complex expressions.
- Example: a−b can be rewritten as a+(−b).
- Distribute Negative Signs in Complex Expressions:
- For expressions with multiple parentheses, distribute negative signs carefully through each set of parentheses.
- Example: For (x+2)−[3(x−4)], distribute the negative sign inside the brackets: (x+2)−[3x−12]=x+2−3x+12.
Subtraction of Algebraic Expressions Solved Examples
Here are five solved examples of subtracting algebraic expressions:
Example 1: Subtract (3x + 2y – 5) from (5x – 2y + 2c).
- Horizontal method: (5x – 2y + 2c) – (3x + 2y – 5) = 5x – 2y + 2c – 3x – 2y + 5 = (5x – 3x) + (-2y – 2y) + (2c + 5) = 2x – 4y + 2c + 5
Example 2:Subtract (4x² – 7x – 5) from (6 + 2x – 3x²).
- Horizontal method: (6 + 2x – 3x²) – (4x² – 7x – 5) = 6 + 2x – 3x² – 4x² + 7x + 5 = (-3x² – 4x²) + (2x + 7x) + (6 + 5) = -7x² + 9x + 11
Example 3:Subtract (4x + 2y – 4z) from (10x – 6y + 2z).
- Horizontal method: (10x – 6y + 2z) – (4x + 2y – 4z) = 10x – 6y + 2z – 4x – 2y + 4z = (10x – 4x) + (-6y – 2y) + (2z + 4z) = 6x – 8y + 6z
Example 4:Subtract (-5ab + 2a²) from (3a² + 9ab).
- Horizontal method: (3a² + 9ab) – (-5ab + 2a²) = 3a² + 9ab + 5ab – 2a² = (3a² – 2a²) + (9ab + 5ab) = a² + 14ab
Example 5:Subtract (2x² – 5xy + 9y² – 4) from (7xy – 2x² – 4y² + 7).
- Horizontal method: (7xy – 2x² – 4y² + 7) – (2x² – 5xy + 9y² – 4) = 7xy – 2x² – 4y² + 7 – 2x² + 5xy – 9y² + 4 = (-2x² – 2x²) + (7xy + 5xy) + (-4y² – 9y²) + (7 + 4) = -4x² + 12xy – 13y² + 11
Also Read: Define Line in Maths: 9 Types and Examples
FAQs
To subtract algebraic expressions, change the sign of every term in the expression being subtracted, then combine like terms. This involves adding the opposite of each term in the subtracted expression to the original expression.
To subtract equations, perform the same operation on both sides of the equation to maintain balance. This involves subtracting the same term or expression from both sides.
5x – 2y is an example of a subtraction expression.
To subtract expressions, change the sign of every term in the expression being subtracted, then combine like terms. For example, (5x + 3) – (2x – 1) becomes 5x + 3 – 2x + 1, which simplifies to 3x + 4.
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