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James has a certain amount of money. If he buys $3$ pens and $2$ pencils, he will have $\$2$ left over. If he buys $2$ pens and $3$ pencils, he will have $\$6$ left over. If Judith arrives with the same amount of money as James, together they can buy $6$ of each and spend all their money. If $a$ is the cost of one pen and $b$ is the cost of one pencil, compute the ordered pair $(a,b)$.

Respuesta :

The cost of pen and pencils are related to the number that can be bought

with a given amount of money.

  • [tex]\underline{The \ ordered \ pair \ (a, \, b) \ is \ (6, \, 2)}[/tex]

Reasons:

The given parameters are;

If  the number number of pens and pencils James buys = 3 pens and 2 pencils, the amount James will have left = $2

If the number of pens and pencils James buys = 2 pens and 3 pencils, the amount James will have left = $6

The amount of money Judith has = The amount of money with James

The number of pens and pencil James and Judith can buy = 6 pens and 6 pencils

The cost of one pen = a

The cost of one pencil = b

Required:

To find the ordered pair (a, b)

Solution:

Let X represent the initial amount of money James has, we get;

X - (3·a + 2·b) = 2...(1)

X - (2·a + 3·b) = 6...(2)

2·X = 6·a + 6·b...(3)

Therefore;

X = (6·a + 6·b) ÷ 2 = 3·a + 3·b

Which from equation (1) gives;

3·a + 3·b - (3·a + 2·b) = 2

3·a + 3·b - 3·a - 2·b = 2

3·b - 2·b = 2

b = 2

Subtracting equation (1) from equation (2) gives;

(X - (2·a + 3·b)) - (X - (3·a + 2·b)) = 6 - 2 = 4

-2·a - 3·b + 3·a + 2·b = 4

a - b = 4

a = 4 + b

∴ a = 4 + 2 = 6

a = 6

[tex]\underline{The \ ordered \ pair \ (a, \, b) \ is \ (6, \, 2)}[/tex]

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