Solving a word problem using a system of linear equations of the form Ax + By = C

Question

A fruit company delivers its fruit in two types of boxes: large and small. A delivery of 2 large boxes and 3 small boxes has a total weight of 77 kilograms. A delivery of 4 large boxes and 8 small boxes has a total weight of 181 kilograms.

How much does each type of box weigh?

Large boxes:

Small boxes:

Answer 1

Hi Kristen

Large Box weighs 18.25 kg

Small Box weighs 13.50 kg

Let L= Large Box

Let S = Small Box

Use the information given to set up equations

Equation 1

2L + 3S = 77

Equation 2

4L + 8S = 181

Multiply Equation 1 by negative and combine it with Equation 2 to eliminate L

-2(2L + 3S = 77)

-4L - 6S = -154

4L + 8S = 181

Combining the equations leaves

2S = 27

Divide both sides by 2

S = 13.5

Substitute 13.5 for S in the original Equation 1 to find L

2L + 3S = 77

2L +3(13.5) = 77

2L + 40.5 = 77

Subtract 40.5 from both sides of the equation

2L = 77 - 40.5

2L =36.5

Divide both sides by 2 to solve for S

L = 36.5/2

L = 18.25 kg

You can check these values in both the equations I hope this helps.

Answer 2

Let x = weight of a single large box

and let y = weight of single small box

Then the word problem tells you:

2x + 3y = 77 (2 large plus 3 small equals 77 kg)

4x + 8y = 181 (4 large plus 8 small equals 181 kg)

Let's solve the first for x:

x = 1/2 * (77 - 3y)

Not put that in for x in the second equation:

2 * (77 - 3y) + 8y = 181

154 - 6y + 8y = 181

2y = 181 - 154

2y = 27

y = 13.5

Not put this into: x = 1/2 * (77 - 3y)

x = 1/2 * (77 - 3 * 13.5) = 1/2 * (77 - 40.5) = 1/2 * (36.5) = 18.25

So the answer is a large box weight 18.25 kg and a small box weighs 13.5 kg.

It's always a good idea to plug these back into the equations to make sure they satisfy them:

4 * (18.25) + 8 (13.5) = 181

73 + 108 = 181

2 * 18.25 + 3 * 13.5 = 77

36.5 + 40.5 = 77

They check