How to Subtract Algebraic Equations?

To subtract algebraic equations, you change the sign of each term in the equation you're subtracting and then add the equations together.

Here's a breakdown of the process with examples:

1. Understanding the Basics

• Like Terms: Remember that you can only add or subtract like terms (terms with the same variable and exponent). For instance, 3x and 5x are like terms, but 3x and 5x² are not.
• Distributing the Negative Sign: Subtracting an expression is the same as adding the negative of that expression. This means you need to distribute the negative sign to every term within the parentheses of the expression being subtracted.

2. Steps for Subtracting Algebraic Equations

1. Write down the equations: Clearly write out the equations you need to subtract, one above the other, aligning like terms in columns if possible.
2. Change the sign: Change the sign of every term in the equation being subtracted. This means:
   • If a term is positive (+), change it to negative (-).
   • If a term is negative (-), change it to positive (+).
3. Add the equations: Add the equations together, combining like terms.

3. Examples

Example 1: Subtracting Simple Expressions

Subtract (2x + 3y) from (5x + 7y).

1. Write down:

   5x + 7y
   - (2x + 3y)

2. Change the sign: Distribute the negative sign to the second expression.

   5x + 7y
   -2x - 3y

3. Add:

   5x + 7y
   -2x - 3y
   ---------
   3x + 4y

   Therefore, (5x + 7y) - (2x + 3y) = 3x + 4y

Example 2: Subtracting with More Terms

Subtract (4a² - 2ab + b²) from (6a² + 5ab - 3b²).

1. Write down:

   6a² + 5ab - 3b²
   - (4a² - 2ab + b²)

2. Change the sign: Distribute the negative sign.

   6a² + 5ab - 3b²
   -4a² + 2ab - b²

3. Add:

   6a² + 5ab - 3b²
   -4a² + 2ab - b²
   ---------
   2a² + 7ab - 4b²

   Therefore, (6a² + 5ab - 3b²) - (4a² - 2ab + b²) = 2a² + 7ab - 4b²

Example 3: Subtracting Equations with a Missing Term

Subtract (x + 5) from (x² + 3x - 2).

1. Write down: It's helpful to align like terms. We can think of the first equation as having a 0x² term.

   x² + 3x - 2
   - (x + 5)

2. Change the sign:

   x² + 3x - 2
   -x - 5

3. Add: Again, aligning like terms helps.

     x² + 3x - 2
   +   -x - 5
   -----------
     x² + 2x - 7

   Therefore, (x² + 3x - 2) - (x + 5) = x² + 2x - 7

4. Key Considerations

• Organization: Keeping your work organized is crucial. Align like terms vertically to minimize errors.
• Careful with Signs: The most common mistake is forgetting to distribute the negative sign to all terms in the expression being subtracted. Double-check each sign.
• Simplifying: After subtracting, always check if you can simplify the resulting expression further by combining any remaining like terms.

By following these steps and practicing regularly, you'll master the process of subtracting algebraic equations.