Select the correct answer.
Which expression is equivalent to the given expression?
$(3 y-4)(2 y+7)+11 y-9$
A. $16 y-6$
B. $6 y^2+11 y+19$
C. $9 y-37$
D. $6 y^2+24 y-37$
Answer (1)
Expand the product: ( 3 y − 4 ) ( 2 y + 7 ) = 6 y 2 + 13 y − 28 .
Add the term 11 y : 6 y 2 + 13 y − 28 + 11 y = 6 y 2 + 24 y − 28 .
Subtract 9: 6 y 2 + 24 y − 28 − 9 = 6 y 2 + 24 y − 37 .
The equivalent expression is 6 y 2 + 24 y − 37 .
Explanation
Understanding the Problem We are given the expression ( 3 y − 4 ) ( 2 y + 7 ) + 11 y − 9 and asked to find an equivalent expression from the given choices. This involves expanding the product, combining like terms, and simplifying the expression.
Expanding the Product First, we expand the product ( 3 y − 4 ) ( 2 y + 7 ) using the distributive property (also known as the FOIL method): ( 3 y − 4 ) ( 2 y + 7 ) = ( 3 y ) ( 2 y ) + ( 3 y ) ( 7 ) + ( − 4 ) ( 2 y ) + ( − 4 ) ( 7 ) = 6 y 2 + 21 y − 8 y − 28 = 6 y 2 + 13 y − 28
Adding the Term 11y Now, we add the term 11 y to the expanded expression: 6 y 2 + 13 y − 28 + 11 y = 6 y 2 + ( 13 y + 11 y ) − 28 = 6 y 2 + 24 y − 28
Subtracting 9 Finally, we subtract 9 from the expression: 6 y 2 + 24 y − 28 − 9 = 6 y 2 + 24 y + ( − 28 − 9 ) = 6 y 2 + 24 y − 37
Final Answer The simplified expression is 6 y 2 + 24 y − 37 . Comparing this with the given options, we find that it matches option D.
Conclusion Therefore, the expression equivalent to ( 3 y − 4 ) ( 2 y + 7 ) + 11 y − 9 is 6 y 2 + 24 y − 37 .
Examples
Understanding how to simplify algebraic expressions is crucial in many real-world applications. For instance, if you're calculating the area of a garden with variable dimensions and then adjusting the design, you'll need to manipulate algebraic expressions to find the new area. Suppose the length of the garden is represented by ( 3 y − 4 ) and the width by ( 2 y + 7 ) . After making adjustments, you need to add an additional 11 y − 9 square units of space. Simplifying the expression ( 3 y − 4 ) ( 2 y + 7 ) + 11 y − 9 allows you to determine the final area in terms of y , which is 6 y 2 + 24 y − 37 . This skill is also fundamental in physics, engineering, and computer science for modeling and solving complex problems.
