Methods to Solve Addition and Subtraction of Algebraic Expressions

• Arrange Like Terms: Write expressions in a way that like terms are aligned vertically.

• Combine Like Terms: Add or subtract the coefficients of like terms.

• Keep Unlike Terms Unchanged: If terms do not have the same variables or exponents, they remain unchanged.

Simplify the Expression: Write the final expression in its simplest form.

Rule for Subtraction as Addition of Additive Inverse

• Subtraction of a number is the same as adding its additive inverse.

• For example, subtracting -3 is the same as adding +3. Similarly:

  • Subtracting 6y is the same as adding -6y.

  • Subtracting -4y² is the same as adding 4y².

• This rule helps simplify expressions by consistently using addition instead of subtraction.

• The signs written below each term in an aligned format indicate the operation to be performed, making calculations clearer.

• Why does this rule work?

  • Mathematically, subtraction can be rewritten as the addition of an opposite number because of the properties of real numbers.

  • The additive inverse of a number is the value that, when added to the original number, results in zero.

  • This means that instead of subtracting a term, we can add its opposite, making calculations easier and eliminating confusion about changing signs.

Multiplication of Algebraic Expressions

• Dot Method to Understand Multiplication of Algebraic Expressions

  • The dot method is a visual way to understand multiplication of algebraic expressions. It represents the multiplication process using rows and columns of dots.

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• Explanation :-

• Breaking Down the Image

  1. Top Section: Basic Multiplication (m × n)

  • The top-left quadrant contains dots arranged in a grid.
  • It consists of m rows and n columns.
  • The total number of dots is found by multiplying the number of rows by the number of columns: $$\text{Total Dots}=m\times n$$
  • The top-right quadrant simply shows that the total number of dots can be written as m × n.

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  2. Bottom Section: Expanding (m + 2) × (n + 3)

  • The bottom-left quadrant introduces an increase in both the number of rows and columns:
    • Rows increase by 2 → New total rows = m + 2.
    • Columns increase by 3 → New total columns = n + 3.
  • This means that we are now finding the total number of dots for: $$(m+2)\times (n+3)$$
  • The bottom-right quadrant visually represents this expanded multiplication.

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  What Does This Mean Algebraically?

  Using the distributive property, we expand:

  $$(m+2)\times (n+3)$$

  Using the distributive property:

  m×n+m×3+2×n+2×3

  This shows that when multiplying two binomials, we:

  1. Multiply the first terms: m × n.
  2. Multiply the outer terms: m × 3.
  3. Multiply the inner terms: 2 × n.
  4. Multiply the last terms: 2 × 3.

  Each of these terms represents a different section of the dot grid.

  Why is the Dot Method Useful?

  • It provides a visual representation of multiplication.
  • It helps understand how the distributive property works.
  • It makes it easier to see how each part of the expression contributes to the final product.

Further we can understand

• Monomial × Monomial → The product is always a monomial (e.g., 2x × 3x = 6x^2).

• Monomial × Polynomial → Multiply the monomial with each term in the polynomial (e.g., 3x(y + 2) = 3xy + 6x).

• Polynomial × Polynomial → Multiply each term of the first polynomial with each term of the second and combine like terms (e.g., (x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6).

• In carrying out the multiplication of a polynomial by a binomial (or trinomial), we multiply term by term, i.e., every term of the polynomial is multiplied by every term in the binomial (or trinomial).Note that in such multiplication, we may get terms in the product which are like and have to be combined.

Real-Life Applications of Multiplication in Algebra

Multiplication of algebraic expressions is widely used in various real-life scenarios. Here are some key applications:

1. Calculating Area and Volume

✅ Finding the Area of a Rectangle

• The area of a rectangle is given by: $$\text{Area}=\text{length}\times \text{breadth}$$
• If the length and breadth are algebraic expressions (e.g., (l + 5) and (b – 3)), the area becomes: $$(l+5)\times (b-3)$$

✅ Finding the Volume of a Box

• The volume of a rectangular box is given by: $$\text{Volume}=\text{length}\times \text{breadth}\times \text{height}$$
• If any of these dimensions change, we use multiplication of polynomials.

2. Speed, Distance, and Time

• Distance traveled is given by: $$\text{Distance}=\text{Speed}\times \text{Time}$$
• If speed and time vary due to external conditions (like traffic or acceleration), we use algebraic expressions to calculate the new distance.

3. Business and Finance

✅ Calculating Total Cost of Items

• If the price per unit of an item is p, and the number of units purchased is n, then: $$\text{Total Cost}=p\times n$$
• If there is a discount or an increase in quantity, we modify the equation using algebra.

✅ Simple Interest Formula

• Interest is calculated using the formula: Interest=Principal×Rate×Time100text{Interest} = frac{text{Principal} times text{Rate} times text{Time}}{100}Interest=100Principal×Rate×Time​
• If the rate or time changes, algebraic expressions help us determine the new interest.

4. Engineering and Construction

• Structural Design: Engineers use multiplication to calculate load-bearing capacity of beams, bridges, and walls.
• Material Estimation: If each brick has a height of h and a wall is n bricks high, the total height is: $$h\times n$$

5. Population Growth and Economics

• If a population grows at a rate of r% per year, after t years, the new population is given by: P×(1+r)tP times (1 + r)^tP×(1+r)t
• This helps in forecasting and resource planning.

6. Science and Physics

✅ Force Calculation

• Newton’s second law states: $$\text{Force}=\text{Mass}\times \text{Acceleration}$$
• If mass or acceleration varies, algebra helps determine the new force.

✅ Chemical Reactions

• The amount of reactants required in a reaction is determined using algebraic multiplication.

7. Shopping and Discounts

• If a shop offers a buy 2, get 1 free offer and the cost per item is p, then: Total cost=p×(Total items purchased3)text{Total cost} = p times left( frac{text{Total items purchased}}{3} right)Total cost=p×(3Total items purchased​)

Conclusion

Multiplication of algebraic expressions is used in everyday calculations, from business transactions to physics equations. It simplifies complex relationships and helps in decision-making in various fields.