The length of a rectangular field is represented by the expression [tex]$14 x-3 x^2+2 y$[/tex]. The width of the field is represented by the expression [tex]$5 x-7 x^2+7 y$[/tex]. How much greater is the length of the field than the width?
A. [tex]$9 x+4 x^2-5 y$[/tex]
B. [tex]$9 x-10 x^2-5 y$[/tex]
C. [tex]$19 x+4 x^2+9 y$[/tex]
D. [tex]$19 x-10 x^2+9 y$[/tex]
Answer (1)
Subtract the expression for the width from the expression for the length: ( 14 x − 3 x 2 + 2 y ) − ( 5 x − 7 x 2 + 7 y ) .
Distribute the negative sign and combine like terms: 14 x − 3 x 2 + 2 y − 5 x + 7 x 2 − 7 y .
Simplify the expression: ( 14 x − 5 x ) + ( − 3 x 2 + 7 x 2 ) + ( 2 y − 7 y ) = 9 x + 4 x 2 − 5 y .
The length is 9 x + 4 x 2 − 5 y greater than the width: 9 x + 4 x 2 − 5 y .
Explanation
Understanding the Problem We are given the length and width of a rectangular field as expressions in terms of x and y . We need to find the difference between the length and the width to determine how much greater the length is than the width.
Setting up the Subtraction The length of the rectangular field is given by the expression 14 x − 3 x 2 + 2 y , and the width is given by the expression 5 x − 7 x 2 + 7 y . To find how much greater the length is than the width, we subtract the width from the length: ( 14 x − 3 x 2 + 2 y ) − ( 5 x − 7 x 2 + 7 y )
Simplifying the Expression Now, we simplify the expression by combining like terms: 14 x − 3 x 2 + 2 y − 5 x + 7 x 2 − 7 y Group the like terms together: ( 14 x − 5 x ) + ( − 3 x 2 + 7 x 2 ) + ( 2 y − 7 y ) Combine the like terms: 9 x + 4 x 2 − 5 y
Final Answer Therefore, the length of the rectangular field is 9 x + 4 x 2 − 5 y greater than the width.
Examples
Understanding how to manipulate and simplify algebraic expressions like this is useful in many real-world scenarios. For example, if you're designing a garden and want to compare the areas of two different rectangular sections, you might use expressions similar to these. By finding the difference between the areas, you can determine how much more space one section has compared to the other, helping you plan your garden layout more effectively. This kind of algebraic thinking is also crucial in fields like engineering, where precise calculations are essential for designing structures and systems.
