A student wants to buy x notebooks and y pens. The student must buy at least 3 notebooks. … — SAT

A student wants to buy $x$ notebooks and $y$ pens. The student must buy at least $3$ notebooks. Each notebook costs $3$ dollars and each pen costs $2$ dollars. If the student can spend a maximum of $15$ dollars, what is the maximum number of pens the student can buy?

Cevap: 3 pens

Cevap

The correct answer is 3. The constraints are represented by the system of inequalities

x \geq 3

and

3x + 2y \leq 15

. To find the maximum number of pens

y

, we minimize the number of notebooks

x

by setting

x = 3

. Substituting

x = 3

into the budget inequality gives

3(3) + 2y \leq 15

, which simplifies to

9 + 2y \leq 15

. Subtracting

9

from both sides results in

2y \leq 6

. Dividing by

2

yields

y \leq 3

. Therefore, the maximum number of pens the student can buy is 3.

Adım Adım Çözüm

Set up the system of inequalities from the given constraints.

The system of inequalities is

x \geq 3

and

3x + 2y \leq 15

, where

x

represents the number of notebooks and

y

represents the number of pens.

To represent the minimum number of notebooks required and the total budget limit.

Substitute the minimum possible value of

x

into the budget inequality to maximize

y

Setting

x = 3

gives

3(3) + 2y \leq 15

, which simplifies to

9 + 2y \leq 15

Since the cost per notebook is positive, minimizing the number of notebooks leaves the maximum amount of budget for purchasing pens.

Solve the inequality for

y

2y \leq 6 \implies y \leq 3

To find the upper bound for the number of pens the student can buy.

Anahtar Kavram

Solving a system of linear inequalities in a real-world context to find a maximum value.

Soru

Cevap

Adım Adım Çözüm

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