Addition Property of Equality

In mathematics, equality is a fundamental concept where two mathematical expressions represent the same quantity. It is represented by the symbol =.

For example, in the equation \(3x + 2x = 5x\); both sides represent the same quantity: 5x.

The addition property of equality states that if the same number is added to both sides of an equation, the equality remains true. In other words, adding the same value to each side of the equation does not affect the equality.

For example, \(x = 4 \)

If we add 3 to both sides, the equation becomes: 

\(x + 3 = 4 + 3 \)

\(x + 3 = 7 \)

For an equation x = y, the addition property of equality states that if the same number n is added to both sides, the equality remains true. 
 

Mathematically it can be represented as: If \(x = y \), then \(x + n = y + n \)

Here x and y can be numbers or algebraic expressions, n is a real number. The addition property of equality is applicable to both arithmetic and algebraic equations. 

For example, \(5 + 7 = 12\), if we add 2 to both sides

Verifying: \(x + n = y + n \)

Here,
 

• \(x = 5 + 7 \)
   
• \(y = 12  \)
   
• \(n = 2 \)

\(5 + 7 + 2 = 12 + 2 \)

Verifying LHS: \(5 + 7 + 2 = 14\)

RHS: \(12 + 2 = 14\)

So, \(LHS = RHS \) 

For example, \(x + 2 = 8 \), adding 5 to both sides

\(x + 2 = 8 \)

Adding 5 on both sides

\(x + 2 + 5 = 8 + 5 \)

\(x + 7 = 13\)

The addition property of equality is also applicable to fractions. In other words, adding the same fraction to both sides of the equation keeps the equation balanced. It can be represented as: 
\(\frac{a}{b} + \frac{x}{y} = \frac{c}{d} + \frac{x}{y} \)

For example, add \(\frac{1}{3}\) to both sides of: \(\frac{3x}{4} = \frac{5}{2} \): 

That is \(\frac{3x}{4} = \frac{5}{2} \)

\(\frac{3x}{4} + \frac{1}{3} = \frac{5}{2} + \frac{1}{3} \)

RHS: \(\frac{5}{2} = \frac{15}{6} \), \(\frac{1}{3} = \frac{2}{6} \), then \(\frac{15}{6} + \frac{2}{6} = \frac{17}{6} \)

LHS: \(\frac{3x}{4} + \frac{1}{3} \)

\(\frac{3x}{4} + \frac{1}{3} \)

\(\frac{3x}{4} = \frac{17}{6} - \frac{1}{3} \)

\(\frac{3x}{4} = \frac{51 - 6}{18} \)

\(3x = \frac{45}{18} \times \frac{4}{1} \)

\(x = \frac{180}{18} = 10 \)

Given below are some important tips that can help students incorporate the addition property of polynomials efficiently while solving problems.
 

• Combine like terms – Only add terms with the same variable and exponent.
  
   
• Arrange neatly – Write polynomials in standard form before adding.
  
   
• Check signs carefully – Keep track of positive and negative coefficients.
  
   
• Use parentheses wisely – Remove them correctly when adding polynomials.
  
   
• Double-check degrees – Ensure the resulting polynomial has terms in descending order of exponents.

We use the addition property of equality in our everyday life for budgeting, cooking, distance calculation. In this section, we will learn a few applications of the addition property of equality. 

• Cooking adjustments: When doubling a recipe, we add equal quantities of each ingredient.
  Example: For 2 cakes \(2 + 2 = 4 \) cups of flour and \(1 + 1 = 2\) cups of sugar.
   
• Solving equations: To find unknowns, we add equal values to both sides.
  Example: \(x - 3 = 7 ⇒ x = 10\) after adding 3 on both sides.
   
• Inventory tracking: When new stock arrives, we add the same number to manual and digital records.
   
• Budget planning: Adding the same expense to both sides keeps income and spending balanced.
   
• Distance calculation: If you travel 5 km more each day, add 5 to both sides to compare total distances.