a local food bank wants to transport as many lunch and clothing packages across the state as it can. each…

a local food bank wants to transport as many lunch and clothing packages across the state as it can. each food package weighs 21 pounds and occupies 5 feet³. each clothing package weighs 19 pounds and occupies 3 feet³. the shipping container holds no more than 5300 pounds and no more than 1100 feet³. write a system of linear inequalities (in slope - intercept form) that describes the constraints of this situation. use the variable y to represent number of food packages and x for number of clothes packages. do not solve.

a local food bank wants to transport as many lunch and clothing packages across the state as it can. each food package weighs 21 pounds and occupies 5 feet³. each clothing package weighs 19 pounds and occupies 3 feet³. the shipping container holds no more than 5300 pounds and no more than 1100 feet³. write a system of linear inequalities (in slope - intercept form) that describes the constraints of this situation. use the variable y to represent number of food packages and x for number of clothes packages. do not solve.

Answer

Explanation:

Step1: Define Variables and Weight Constraint

Let ( y ) = number of food packages, ( x ) = number of clothing packages.
Weight constraint: ( 21y + 19x \leq 5300 ).
Solve for ( y ):
( 21y \leq -19x + 5300 )
( y \leq -\frac{19}{21}x + \frac{5300}{21} ).

Step2: Define Volume Constraint

Volume constraint: ( 5y + 3x \leq 1100 ).
Solve for ( y ):
( 5y \leq -3x + 1100 )
( y \leq -\frac{3}{5}x + 220 ).

Step3: Non - negativity Constraints

Since the number of packages can't be negative:
( y \geq 0 ) (or ( y \leq \infty ) but non - negative, so ( y \geq 0 ) can be written as ( y \leq \infty ) but in context, ( y \geq 0 ) and ( x \geq 0 ). For slope - intercept, ( y \geq 0 ) is ( y \leq \infty ) (not useful) but ( x \geq 0 ) and ( y \geq 0 ) are implicit, but the main inequalities from weight and volume are the first two.

Answer:

The system of linear inequalities is:
( y \leq -\frac{19}{21}x + \frac{5300}{21} )
( y \leq -\frac{3}{5}x + 220 )
( x \geq 0 )
( y \geq 0 ) (The first two are the main constraint inequalities in slope - intercept form from weight and volume)