Question

An auto manufacturer sends cars from its two plants, Plant I and Plant II, to two delearships, dealership A and delearship B. Plant I ships 85 cars to the delearships and Plant II ships 70 cars. Dealership A needs 53 cars and dealership B needs 102 cars. The cost for shipping a car from Plant I to dealerships A and B is $310 and $220, respectively. The cost for shipping a car from Plant II to dealerships A and B is $330 and $150, respectively. There are $35050 available for shipping the cars.

Let x represent the number of cars shipped from Plant I to dealership A. Let y represent the number of cars shipped from Plant II to dealership A. Let z represent the number of cars shipped from Plant I to dealership B. Let w represent the number of cars shipped from Plant II to dealership B.

An auto manufacturer sends cars from its two plants, Plant I and Plant II, to two delearships, dealership A and delearship B. Plant I ships 85 cars to the delearships and Plant II ships 70 cars. Dealership A needs 53 cars and dealership B needs 102 cars. The cost for shipping a car from Plant I to dealerships A and B is $310 and $220, respectively. The cost for shipping a car from Plant II to dealerships A and B is $330 and $150, respectively. There are $35050 available for shipping the cars. Let & represent the number of cars shipped from Plant I to dealership A. Let y represent the number of cars shipped from Plant II to dealership A. Let z represent the number of cars shipped from Plant I to dealership B. Let w represent the number of cars shipped from Plant Il to dealership B. Find an equation for the total number of cars shipped to dealership A: x+y=53 Find an equation for the total number of cars shipped to dealership B: 2+w=102 Find an equation for the total number of cars shipped from Plant 1: *+2=85 Find an equation for the total number of cars shipped from Plant II: y+w=70 Find an equation for the total cost of shipping the cars from the two plants to the two dealerships: 310x + 220z+3304+ 150w=35050 Use those equations to find the following. The number of cars shipped from Plant I to dealership A: The number of cars shipped from Plant II to dealership A: The number of cars shipped from Plant I to dealership B: The number of cars shipped from Plant II to dealership B:

Answer 1

Consider the equations

$x+y=53$    equation 1      

$z+w=102$ equation 2

$x+z=85$ equation 3

equation4

y +w= 70

3100 + 2202 + 330y + 150w = 35050
equation5

(Equation 4)-(equation 1)

$\Rightarrow y+w-(x+y)=70-53$

$\Rightarrow y+w-x-y=70-53$

$w-x=17$

$w=17+x$ equation a

(equation 3)-(equation1)

>I+2 - (2 + y) = 85 – 53

$\Rightarrow x+z-x-y=85-53$

$\Rightarrow z-y=32$

$z=32+y$ equation b

consider equation 5

$\Rightarrow 310x+220z+330y+150w=35050$

$\Rightarrow 31x+22z+33y+15w=3505$

substitute $w=17+x$ and $z=32+y$

$\Rightarrow 31x+22(32+y)+33y+15(17+x)=3505$

$\Rightarrow 31x+704+22y+33y+255+15x=3505$

$\Rightarrow 46x+55y=3505-(704+255)$

$\Rightarrow 46x+55y=2546$ equation 6

consider equatioequn 1

$x+y=53$

multiplying this equation by 46 gives

$46x+46y=53*46$

$46x+46y=2438$ equation 7

(equation 6)-(equation 7)

$46x+55y-46x-46y=2546-2438$

$55y-46y=108$

$9y=108$

$\Rightarrow y=12$

y is the number of cars shipped from plant II to dealership A

$\therefore\;12\;cars\;are\;shipped\;from\;plantII\;to\;dealership\;A$

Substitute y=12 in equation 1

x+y=53

x+12=53

x=53-12

x=41

x is the number of cars shipped from plant I to dealership A

Therefore 41 cars are shipped from plant I to dealership A

Substitute x=41 in equation 3

x+z=85

41+z=85

z=85-41

z=44

z represent number of cars shipped from plant I to dealership B

Therefore 44 cars are shipped from plant I to dealership B

Substitute z=44 in equation 2

z+w=102

44+w=102

w=102-44

w=58

Number of cars shipped from plant II to dealership B is 58