Subtraction of Algebraic Expressions

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Subtraction of Algebraic Expressions

Subtraction of algebraic expressions is similar to adding them, but with a slight twist. To subtract algebraic expressions, you need to combine like terms and perform the subtraction operation following the rules of algebra. Here’s a step-by-step guide:

Identify Like Terms: Like terms are terms that have the same variables raised to the same powers. For example, in the expressions 3x^2 + 2x – 5 and 2x^2 – 4x + 7, the like terms are the terms with ‘x^2’ and ‘x’. Terms without the same variables or powers are not like terms.

Arrange the Expressions: Line up the like terms of both expressions vertically, keeping the corresponding terms aligned.

Perform Subtraction: Subtract the coefficients of like terms. Keep the variable and exponent the same. If there’s no corresponding term in one of the expressions, you can subtract zero.

Combine Unlike Terms: After subtracting the like terms, rewrite the rest of the terms that are not like terms as they are. These terms cannot be simplified further.

Simplify (if possible): Combine any constants that result from the subtraction of like terms.

Final Expression: Write down the simplified expression, combining the simplified terms.

Here’s an example to illustrate:

Subtract: 3x^2 + 2x – 5 from 2x^2 – 4x + 7

Step 1: Identify Like Terms

Like terms with ‘x^2’: 3x^2 from the first expression, 2x^2 from the second expression.

Like terms with ‘x’: 2x from the first expression, -4x from the second expression.

Step 2: Arrange the Expressions

2x^2 – 4x + 7

– 3x^2 + 2x – 5

Step 3: Perform Subtraction

(2x^2 – 3x^2) + (-4x + 2x) + 7 – (-5)

Step 4: Combine Unlike Terms

-x^2 – 2x + 12

Step 5: Simplify

The expression is already simplified in this case.

So, the result of subtracting 3x^2 + 2x – 5 from 2x^2 – 4x + 7 is:

-x^2 – 2x + 12

Always remember to carefully follow the rules of algebra when subtracting algebraic expressions to ensure accurate results.

What is the Definition of Subtraction of Algebraic Expressions?

Subtraction of algebraic expressions involves the process of combining or subtracting two or more algebraic terms or expressions. Algebraic expressions are mathematical representations that consist of variables, constants, and mathematical operations like addition, subtraction, multiplication, and division. When subtracting algebraic expressions, you need to carefully handle the signs and coefficients associated with each term.

The general process of subtracting algebraic expressions involves the following steps:

Identify like terms: Like terms are terms that have the same variables raised to the same powers. For example, in the expressions 3x^2 + 2x and -5x^2 + 3x, the like terms are the terms with x^2 and x. Identify and group these like terms together.

Distribute the negative sign: When subtracting an expression, it’s common to distribute the negative sign to all the terms within the expression that you’re subtracting. This is done by changing the sign of each term in the expression being subtracted.

Combine like terms: After distributing the negative sign and changing the signs of the terms in the expression being subtracted, you can combine the like terms with the terms in the original expression.

Simplify the result: Finally, simplify the resulting expression by combining any like terms that might have arisen during the subtraction process.

How can we Subtract Algebraic Expressions?

Subtracting algebraic expressions involves simplifying and combining like terms in a similar way to adding them. Here’s a step-by-step process to subtract algebraic expressions:

Let’s say you have two algebraic expressions A and B that you want to subtract: A – B.

Identify Like Terms: Before you begin subtracting, identify the like terms in both expressions. Like terms are terms that have the same variable(s) raised to the same power(s). For example, 3x and 2x are like terms, but 3x^2 and 2x are not.

Distribute Negative Sign (if necessary): If you’re subtracting a whole expression, distribute the negative sign across all the terms in the second expression (B). This involves changing the sign of each term in B. For example, if B = 2x – 4y, then distribute the negative sign to get -2x + 4y.

Combine Like Terms: Now combine the like terms from both expressions. For each variable, add or subtract the coefficients. If a term is present in one expression but not the other, just copy it over.

Simplify: After combining like terms, simplify the expression by further adding or subtracting any constants.

Here’s an example:

Let’s subtract the expressions: A = 3x^2 – 2xy + 5y and B = x^2 + 3xy – 2y.

Identify like terms:

Like terms with x: 3x^2 and x^2.

Like terms with y: -2xy and 3xy.

Constants: 5y and -2y.

Distribute the negative sign:

B = -x^2 – 3xy + 2y.

Combine like terms:

x^2 – x^2 = 0 (eliminate the x^2 terms).

-2xy + 3xy = xy.

5y + 2y = 7y.

Simplify:

Final expression: 0 + xy + 7y, which can be written as: xy + 7y.

So, A – B = xy + 7y.

Always remember to carefully distribute the negative sign and combine like terms accurately to get the correct result when subtracting algebraic expressions.

Horizontal Method for Subtraction of Algebraic Expressions

The horizontal method for subtracting algebraic expressions is a technique used to simplify and perform subtraction operations on algebraic expressions. This method is similar to the traditional vertical subtraction method used for numbers. It’s particularly helpful when dealing with larger and more complex expressions. Here’s how you can use the horizontal method for subtracting algebraic expressions:

Let’s consider the subtraction of two algebraic expressions: A – B.

Write down the two expressions, A and B, side by side with their like terms aligned vertically. Make sure to arrange the terms in descending order of their exponents.

A: 3x^2 + 2x – 5

B: 2x^2 – 4x + 7

Perform the subtraction term by term. Subtract each corresponding term of expression B from expression A. Remember to change the sign of each term in expression B before performing the subtraction.

A – B: (3x^2 – 2x^2) + (2x + 4x) – (5 + 7)

= x^2 + 6x – 12

Write down the simplified result after performing the subtraction.

Result: x^2 + 6x – 12

In this method, it’s important to keep track of the signs and carefully subtract the corresponding terms. You might need to combine like terms and simplify the final result further if possible.

Remember to maintain proper alignment and organization while using the horizontal method to ensure accuracy in your calculations. This technique can be particularly useful when dealing with larger expressions that have numerous terms.

Column Method for Subtraction of Algebraic Expressions

The column method for subtraction of algebraic expressions is a technique used to subtract two or more algebraic expressions in a structured and organized manner. It’s similar to the traditional column method used for subtracting numbers. This method is particularly helpful when dealing with complex algebraic expressions involving multiple terms.

Here’s a step-by-step guide on how to use the column method for subtracting algebraic expressions:

Let’s consider an example: (3x^2 + 2x – 5) – (x^2 – 4x + 7)

Step 1: Align the expressions vertically with like terms in the same columns.

3x^2 + 2x – 5

– x^2 – 4x + 7

Step 2: Begin by subtracting the terms in the first column (terms with the same exponent).

Subtract the coefficients of the terms with the same exponent:

3x^2 – x^2 = 2x^2

Step 3: Move to the second column and perform the subtraction for the terms with the same exponent.

Subtract the coefficients of the terms with the same exponent:

2x – (-4x) = 2x + 4x = 6x

Step 4: Proceed to the third column and subtract the constant terms.

Subtract the constants:

-5 – 7 = -12

Step 5: Write the final result by combining the terms.

The result of the subtraction is:

2x^2 + 6x – 12

So, (3x^2 + 2x – 5) – (x^2 – 4x + 7) simplifies to 2x^2 + 6x – 12.

Remember to be careful with signs and exponents during each step of the process.

This method can also be extended to subtract more than two expressions by aligning their corresponding terms in the same columns and following the same process for each column.

What is an Example of Subtraction of Algebraic Expressions?

Here’s an example of subtracting algebraic expressions:

Let’s say we have two algebraic expressions:

• Expression 1: 3x^2 + 5x – 2
• Expression 2: 2x^2 – 4x + 7

To subtract Expression 2 from Expression 1, we’ll subtract the corresponding coefficients of like terms. Like terms are terms that have the same variable and exponent combination.

Let’s perform the subtraction step by step:

Expression 1: 3x^2 + 5x – 2

Expression 2: -(2x^2 – 4x + 7) (Note: Subtracting an expression is the same as adding its negation)

Distribute the negative sign across Expression 2:

Expression 2: -2x^2 + 4x – 7

Now, subtract the corresponding coefficients of like terms:

(3x^2 – 2x^2) + (5x – 4x) – (-2 – 7)

Simplify each part:

x^2 + x – (-9)

Simplify the double negative:

x^2 + x + 9

So, the result of subtracting Expression 2 from Expression 1 is:

x^2 + x + 9

Some Solved Examples of Subtraction of Algebraic Expressions

Here are a few examples of subtracting algebraic expressions along with their solutions:

Example 1:

Simplify the expression: 3x² – (2x² + 5x – 1)

Solution:

Distribute the negative sign to all terms within the parentheses:

3x² – 2x² – 5x + 1

Combine like terms:

(3x² – 2x²) – 5x + 1 = x² – 5x + 1

Example 2:

Simplify the expression: (4y² – 7y + 3) – (y² + 2y – 5)

Solution:

Distribute the negative sign to all terms within the second parentheses:

4y² – 7y + 3 – y² – 2y + 5

Combine like terms:

(4y² – y²) – (7y + 2y) + (3 + 5) = 3y² – 9y + 8

Example 3:

Simplify the expression: (2a³ – 5a² + 3a – 1) – (a³ – 2a² + 4a + 2)

Solution:

Distribute the negative sign to all terms within the second parentheses:

2a³ – 5a² + 3a – 1 – a³ + 2a² – 4a – 2

Combine like terms:

(2a³ – a³) – (5a² – 2a²) + (3a – 4a) + (-1 – 2) = a³ – 3a² – a – 3

These examples demonstrate how to subtract algebraic expressions by distributing the negative sign and then combining like terms. The key is to pay attention to signs and perform the operations step by step.

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