How Do You Find Equivalent Pairs?

You find equivalent pairs by verifying if they represent the same value, often by manipulating one or both elements of the pair to a common form and comparing. This applies to various mathematical contexts like fractions, ratios, or expressions.

Equivalent Fractions

The most common example is equivalent fractions. Here's how to determine if two fractions are equivalent:

1. Simplifying: Reduce both fractions to their simplest form. If the simplified fractions are the same, then the original fractions are equivalent.

2. Common Denominator: Find a common denominator for both fractions. Then, adjust the numerators accordingly. If the numerators are the same after the denominators are equal, the fractions are equivalent. This leverages the principle that multiplying the numerator and denominator of a fraction by the same non-zero number doesn't change its value.

   • Example: Are 2/3 and 6/9 equivalent?
     • Find a common denominator: The least common multiple of 3 and 9 is 9.
     • Convert 2/3 to have a denominator of 9: (2/3) * (3/3) = 6/9
     • Compare: 6/9 = 6/9. Therefore, 2/3 and 6/9 are equivalent.

3. Cross-Multiplication: Multiply the numerator of the first fraction by the denominator of the second fraction and vice versa. If the two products are equal, the fractions are equivalent.

   • Example: Are 2/3 and 6/9 equivalent?
     • Cross-multiply: 2 9 = 18 and 3 6 = 18
     • Compare: 18 = 18. Therefore, 2/3 and 6/9 are equivalent.

Equivalent Ratios

Ratios can also be equivalent. The process is similar to fractions:

1. Simplifying: Simplify each ratio by dividing both sides by their greatest common factor.

2. Scaling: Check if one ratio can be multiplied by a constant to obtain the other ratio.

   • Example: Are the ratios 4:6 and 6:9 equivalent?
     • Simplify: 4:6 simplifies to 2:3. 6:9 simplifies to 2:3.
     • Compare: Both simplify to 2:3. Therefore, 4:6 and 6:9 are equivalent.

Equivalent Expressions

In algebra, expressions are equivalent if they produce the same result for all possible values of the variable(s).

1. Simplification: Simplify both expressions as much as possible.

2. Substitution: Substitute different values for the variable(s) into both expressions. If the results are the same for all tested values, the expressions are likely equivalent. (However, this isn't a definitive proof; it only increases confidence).

3. Algebraic Manipulation: Use algebraic identities and properties to transform one expression into the other.

   • Example: Are the expressions 2(x + 3) and 2x + 6 equivalent?
     • Distribute: 2(x + 3) = 2x + 6
     • Compare: 2x + 6 = 2x + 6. Therefore, the expressions are equivalent.

Summary

Finding equivalent pairs involves simplifying, manipulating, and comparing the elements to determine if they represent the same underlying value. The specific method depends on the type of "pair" you're working with (fractions, ratios, expressions, etc.).