To determine which property justifies the statement [tex]\(5(7z - 3y) = 35z - 15y\)[/tex], let’s examine the options given:
1. Reflexive Property of Equality: This property states that any mathematical expression is equal to itself. For example, [tex]\(a = a\)[/tex]. This does not apply to our equation because we are transforming one expression into another.
2. Symmetric Property of Equality: This property states that if one quantity equals another, then they can be written in reverse order, such as if [tex]\(a = b\)[/tex] then [tex]\(b = a\)[/tex]. This property also does not apply to our equation, as we are not merely swapping the sides of an equation.
3. Transitive Property of Equality: This property states that if [tex]\(a = b\)[/tex] and [tex]\(b = c\)[/tex], then [tex]\(a = c\)[/tex]. This does not apply to our equation either, because we are not proving a sequence of equalities.
4. Distributive Property: This property states that [tex]\(a(b + c) = ab + ac\)[/tex] and is used to distribute a factor over terms inside parentheses. In our case, the distributive property can be applied as follows:
[tex]\[
5(7z - 3y) = (5 \cdot 7z) + (5 \cdot -3y) = 35z - 15y
\][/tex]
Therefore, the property that justifies the statement [tex]\(5(7z - 3y) = 35z - 15y\)[/tex] is the Distributive Property.
