Problem 6 - Algebra A Combined Function

Question

Samuel has 28 gallons of saline solution available in two large containers at his pharmacy. One container holds three times as much as the other container. Find the capacity of each container. a. 15 gallons; 5 gallons b. 20 gallons; 8 gallons c. 21 gallons; 7 gallons

Step-by-Step Solution

Answer

21 gallons; 7 gallons

Step 1: Define Variables

Let's define two variables. Let \( x \) represent the gallons in the smaller container. Since the larger container holds three times as much as the smaller one, it can be represented by \( 3x \).

Step 2: Set Up the Equation

According to the problem, the total volume of the two containers is 28 gallons. Thus, we have the equation: \( x + 3x = 28 \).

Step 3: Combine Like Terms

Combine the terms on the left side of the equation: \( 4x = 28 \).

Step 4: Solve for x

Divide both sides of the equation by 4 to isolate \( x \): \[ x = \frac{28}{4} = 7 \].

Step 5: Determine Capacities

Now that we know \( x = 7 \), the smaller container has 7 gallons, and the larger one has three times this amount: \( 3 \times 7 = 21 \) gallons.

Key Concepts

Defining Variables

The very first step in solving any algebra word problem involves defining the variables. This means identifying what quantities in the problem are unknown and assigning them symbols to make the problem easier to manage.
In our exercise, we begin by focusing on what we don't know - the capacity of the two containers. We define the smaller container's capacity as \( x \) gallons.
The larger container's capacity is mentioned as being three times the smaller container's, which allows us to express it as \( 3x \).
This systematic naming of variables helps simplify our equations later on, allowing us to focus on solving a concrete problem rather than getting lost in the words of the exercise.

Setting Up the Equation

Once you've defined your variables, the next crucial step is setting up an equation that represents the scenario described in the problem.
This means translating the words of the problem into a mathematical statement using your variables.
In our saline solution exercise, we know that the combined volume of both containers is 28 gallons.
So, we create the equation: \( x + 3x = 28 \). Here \( x \) is the smaller container's capacity, and \( 3x \) is the larger container's. It's important to ensure that this equation faithfully represents the relationships described in the problem, as it's essentially the core of the solution.

Solving Equations

With your equation set up, it's time to solve it. Solving equations typically involves finding the value of the unknown variable that makes the equation true. This process might require different techniques depending on the complexity of the equation.
In our current problem, we focus on isolating \( x \) to find its specific value. By combining like terms, which we'll discuss next, we arrive at the simpler equation: \( 4x = 28 \).
To solve for \( x \), we perform basic algebraic manipulation by dividing both sides by 4, giving us \( \frac{28}{4} \). This results in \( x = 7 \), meaning the smaller container holds 7 gallons.

Combining Like Terms

An important skill in solving equations is the ability to combine like terms. Like terms are terms that have the same variables raised to the same power, and they can be added or subtracted from each other.
In our equation \( x + 3x = 28 \), \( x \) and \( 3x \) are like terms because they both involve \( x \) to the first power.
Combining them simplifies the equation to \( 4x = 28 \). This step not only makes the equation easier to solve but also reduces the number of steps to reach the solution.
This process is crucial in making complicated expressions simpler and more manageable to solve.