At a community farm, strawberries are packed in small crates that hold 3 kilograms each, and watermelons are packed in large crates that hold 7 kilograms each. A shipment contains a combined total of 116 kilograms of strawberries and watermelons. If represents the number of small crates of strawberries and represents the number of large crates of watermelons, the relationship between and can be written as If both and must be whole numbers, what is the greatest possible value of ?

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SAT Linear Equations in Two Variables Question 42 | Free SAT Prep | Aniko

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Question 42·Hard·Linear Equations in Two Variables

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At a community farm, strawberries are packed in small crates that hold 3 kilograms each, and watermelons are packed in large crates that hold 7 kilograms each. A shipment contains a combined total of 116 kilograms of strawberries and watermelons. If $x$ represents the number of small crates of strawberries and $y$ represents the number of large crates of watermelons, the relationship between $x$ and $y$ can be written as

3x + 7y = 116.

If both $x$ and $y$ must be whole numbers, what is the greatest possible value of $y$ ?

For linear word problems with an equation and whole-number conditions, first translate the story into an equation (often already given), then solve for one variable in terms of the other. Use the fact that the result must be a whole number to test only the answer choices, starting from the largest when the problem asks for a maximum. Check each candidate quickly by substitution and divisibility rather than solving the equation from scratch for every choice, which saves time and reduces mistakes.

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Focus on the equation and the variables

You are given $3x + 7y = 116$ . Think about how $x$ and $y$ are related if this equation must be true and both must be whole numbers.

Solve for one variable in terms of the other

Try isolating $x$ in the equation so it is written in the form $x = \text{(expression with } y\text{)}$ . This will help you check which $y$ values make $x$ a whole number.

Use the answer choices strategically

Since the question asks for the greatest possible value of $y$ , start testing the given answer choices from the largest value of $y$ downwards and see when $x$ becomes a whole number.

Check divisibility

When you plug in a value for $y$ , make sure $116 - 7y$ is divisible by 3 so that $x = \frac{116 - 7y}{3}$ is a whole number.

Desmos Guide

Set up the expression for x in terms of y

In Desmos, type x = (116 - 7y)/3. This defines $x$ as a function of $y$ so you can quickly see whether $x$ is a whole number for different values of $y$ .

Use a table to test the answer choices

Create a table in Desmos with a column for $y$ , and enter the four values 11, 12, 13, and 14. In the $x$ column of the table, use the expression (116 - 7y)/3 so Desmos computes the corresponding $x$ values for each $y$ .

Identify the greatest valid y

Look down the $x$ column and see which rows give $x$ as a whole number (no decimal part). Among those rows, find the largest $y$ value; that $y$ is the answer.

Step-by-step Explanation

Understand what the equation represents

The equation

3x + 7y = 116

comes from the total weight:

Each small crate of strawberries weighs 3 kg, so $x$ small crates weigh $3x$ kg.
Each large crate of watermelons weighs 7 kg, so $y$ large crates weigh $7y$ kg. Their combined weight is 116 kg. We are told $x$ and $y$ must be whole numbers (nonnegative integers), and we want the greatest possible value of $y$ from the answer choices.

Express one variable in terms of the other

Solve the equation for $x$ in terms of $y$ :

3x + 7y = 116 \\[0.7em] 3x = 116 - 7y \\[0.7em] x = \frac{116 - 7y}{3}

For a given value of $y$ , $x$ will be a whole number only if $116 - 7y$ is a multiple of 3 and is not negative.

Use the answer choices and test from largest to smallest

Because we want the greatest possible value of $y$ , it is efficient to test the choices from largest to smallest.

Try $y = 14$ :

x = \frac{116 - 7(14)}{3} = \frac{116 - 98}{3} = \frac{18}{3} = 6

Here, $x = 6$ , which is a whole number.

Try $y = 13$ :

x = \frac{116 - 7(13)}{3} = \frac{116 - 91}{3} = \frac{25}{3}

$\frac{25}{3}$ is not a whole number.

Try $y = 12$ :

x = \frac{116 - 7(12)}{3} = \frac{116 - 84}{3} = \frac{32}{3}

$\frac{32}{3}$ is not a whole number.

Try $y = 11$ :

x = \frac{116 - 7(11)}{3} = \frac{116 - 77}{3} = \frac{39}{3} = 13

Here, $x = 13$ , which is a whole number.

So both $y = 14$ and $y = 11$ give whole-number values for $x$ , but we need the greatest possible $y$ .

Choose the greatest valid value of y

Among the answer choices, the values of $y$ that make $x$ a whole number are 14 and 11. Since we want the greatest possible value of $y$ , the correct choice is $y = 14$ .

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation