Let's analyze the statement provided: "If [tex]\( x = 2 \)[/tex], and [tex]\( y = 3x - 11 \)[/tex], then [tex]\( y = 3 \cdot 2 - 11 \)[/tex]."
We need to determine which mathematical property justifies replacing [tex]\( x \)[/tex] with [tex]\( 2 \)[/tex] in the expression for [tex]\( y \)[/tex].
Consider each of the provided options:
1. Addition Property of Equality:
This property states that if [tex]\( a = b \)[/tex], then [tex]\( a + c = b + c \)[/tex]. It involves adding the same value to both sides of an equation and does not directly relate to our given transformation.
2. Division Property of Equality:
This property indicates that if [tex]\( a = b \)[/tex], then [tex]\( \frac{a}{c} = \frac{b}{c} \)[/tex], provided [tex]\( c \neq 0 \)[/tex]. It involves dividing both sides of an equation by the same non-zero number and is not related to our given transformation.
3. Subtraction Property of Equality:
This property states that if [tex]\( a = b \)[/tex], then [tex]\( a - c = b - c \)[/tex]. Similar to the Addition Property, this involves subtracting the same value from both sides of an equation and does not directly correspond to our scenario.
4. Substitution Property of Equality:
This property asserts that if [tex]\( a = b \)[/tex], then [tex]\( b \)[/tex] can be substituted for [tex]\( a \)[/tex] in any expression. It means that any instance of [tex]\( a \)[/tex] in an expression can be replaced with [tex]\( b \)[/tex] without changing the equality.
In our case:
- We start with [tex]\( x = 2 \)[/tex].
- The expression for [tex]\( y \)[/tex] is given as [tex]\( y = 3x - 11 \)[/tex].
- By substituting [tex]\( 2 \)[/tex] for [tex]\( x \)[/tex], we transform the expression to [tex]\( y = 3 \cdot 2 - 11 \)[/tex].
This is a direct application of the Substitution Property of Equality because we are replacing [tex]\( x \)[/tex] with its known value [tex]\( 2 \)[/tex] in the expression for [tex]\( y \)[/tex].
Therefore, the correct property that justifies the statement is the Substitution Property of Equality.
