Equivalent expressions and the distributive property Knowledge-Based Study

Study notes

1. What Are Equivalent Expressions?

   Two algebraic expressions are called equivalent expressions if they always have the same value, no matter what number is substituted for the variable.

   For example, consider the expressions:

   • Expression 1: $x + x + x$
   • Expression 2: $3x$

   If we substitute $x = 4$:

   • $4 + 4 + 4 = 12$
   • $3 \times 4 = 12$

   Both expressions evaluate to 12. Because this equality holds true for every possible value of $x$, we can write:

   $$x + x + x = 3x$$

   Testing specific numbers can help you check whether two expressions might be equivalent, but algebraic properties allow us to prove they are equivalent for all values.

   Expanding with the Distributive Property

   The distributive property states that multiplying a sum or difference by a number gives the same result as multiplying each term individually and then adding or subtracting the products.

   For any numbers or algebraic terms $a$, $b$, and $c$:

   $$a(b + c) = ab + ac$$$$a(b - c) = ab - ac$$

   Example

   To rewrite $4(2x + 5)$ without parentheses, distribute the factor 4 to each term inside:

   • Multiply 4 by $2x$: $4 \times 2x = 8x$
   • Multiply 4 by 5: $4 \times 5 = 20$

   Combining these products gives the equivalent expression:

   $$4(2x + 5) = 8x + 20$$

   Careful with Subtraction: Always keep track of signs. For instance, $3(4x - 7) = (3 \times 4x) - (3 \times 7) = 12x - 21$.

   Reverse Distribution: Factoring

   We can also apply the distributive property in reverse. When an expression is written as a sum or difference, we can factor out the greatest common factor (GCF) of the terms.

   $$ab + ac = a(b + c)$$

   Example

   Consider the expression $18x + 24$:

   1. Find the greatest common factor of the numerical coefficients 18 and 24, which is 6.
   2. Rewrite each term as a product involving 6:
      • $18x = 6 \times 3x$
      • $24 = 6 \times 4$
   3. Factor out the 6 using the distributive property:

   $$18x + 24 = 6(3x + 4)$$

   Both $18x + 24$ and $6(3x + 4)$ are equivalent expressions representing the exact same quantity.

   Simplifying and Proving Equivalence

   Often, determining whether two expressions are equivalent requires both using the distributive property and combining like terms.

   Example: Checking Equivalence

   Are the expressions $2(3x + 4) + 5x$ and $11x + 8$ equivalent?

   1. Distribute the 2 across $(3x + 4)$:
      $2(3x + 4) + 5x = 6x + 8 + 5x$
   2. Combine the like terms $6x$ and $5x$:
      $(6x + 5x) + 8 = 11x + 8$

   Since the simplified form is $11x + 8$, the two expressions are indeed equivalent.

   Common Misconceptions

   • Incomplete Distribution: Multiplying only the first term inside the parentheses (e.g., incorrectly writing $5(x + 2)$ as $5x + 2$ instead of $5x + 10$). Every term inside the parentheses must be multiplied.
   • Sign Errors with Negatives: When distributing a negative value, such as $-3(2x - 4)$, remember that a negative times a negative is positive:
     $$-3(2x - 4) = -6x + 12$$