Identify the correct logical reason for the next step in solving this equation:
[tex]3(2-3 x)+4 x=x-7[/tex]
A. Multiplication Property of Equality
B. Addition Property of Equality
C. Distributive Property of Equality
D. Subtraction Property of Equality
Answer (1)
Apply the distributive property to expand 3 ( 2 − 3 x ) to 6 − 9 x .
The equation becomes 6 − 9 x + 4 x = x − 7 .
The correct property is the Distributive Property of Equality.
The final answer is Distributive Property of Equality.
Explanation
Analyze the equation The given equation is 3 ( 2 − 3 x ) + 4 x = x − 7 . To solve this equation, we need to simplify it step by step. The first step involves expanding the left side of the equation by applying the distributive property.
Apply the distributive property The distributive property states that a ( b + c ) = ab + a c . In our equation, we have 3 ( 2 − 3 x ) , which can be expanded using the distributive property as 3 × 2 − 3 × 3 x = 6 − 9 x .
Simplify the equation After applying the distributive property, the equation becomes 6 − 9 x + 4 x = x − 7 . The next step would be to combine like terms on the left side of the equation.
Identify the correct property Therefore, the correct logical reason for the next step in solving the equation is the Distributive Property of Equality.
Examples
The distributive property is used in everyday life, for example, when calculating the total cost of multiple items. If you buy 3 items that cost $2 each and also have to pay $1 in tax, the total cost can be calculated as $3(2+1) = 3 \times 2 + 3 \times 1 = 6 + 3 = $9. This is an application of the distributive property.
