Problem 23 - Beginning and Intermediate Algebra

Question

Americans own a total of about 355 million fish, cats, and dogs as pets. The number of fish owned is 11 million more than the total number of cats and dogs owned, and 16 million more cats are owned than dogs. (Source: American Pet Products Manufacturers Association) How many of each type of pet do Americans own?

Step-by-Step Solution

Answer

Americans own 183 million fish, 94 million cats, and 78 million dogs as pets.

Step 1: Define the variables

Let's define the variables for the number of each type of pet: - x: number of fish owned - y: number of cats owned - z: number of dogs owned

Step 2: Set up the equations

We are given three pieces of information that can be translated into equations as follows: 1. The total number of fish, cats, and dogs is 355 million: \(x + y + z = 355\) 2. The number of fish owned is 11 million more than the total number of cats and dogs owned: \(x = y + z + 11\) 3. 16 million more cats are owned than dogs: \(y = z + 16\)

Step 3: Solve the system of equations

We have three equations with three variables. To solve this system, we can use the substitution or elimination method. Let's use the substitution method: From equation (3), we know that \(y=z+16\). Plug this into equation (2) to eliminate y: \(x = (z + 16) + z + 11\) Now, we have an equation with only two variables:

Step 4: Simplify the equation

Simplify the equation to solve for x in terms of z: \(x = 2z + 27\)

Step 5: Substitute the equation into the first equation

Now, we can plug in this expression for x into the first equation to find a relationship between y and z: \((2z + 27) + y + z = 355\)

Step 6: Substitute the equation for y

Now, substitute the equation for y in terms of z to the equation above: \((2z + 27) + (z + 16) + z = 355\)

Step 7: Solve for z

Simplify and solve this equation for z: \(4z + 43 = 355\) \(4z = 312\) \(z = 78\) Now we know that Americans own 78 million dogs.

Step 8: Solve for y and x

Now, using the values of z, we can find y and x using equations (3) and (4): \(y = z + 16 = 78 + 16 = 94\) \(x = 2z + 27 = 2(78) + 27 = 156 + 27 = 183\)

Step 9: Check the solution

Let's check if our solution is correct by plugging in the values of x, y, and z into the original equations: 1. \(x + y + z = 355 = 183 + 94 + 78 = 355\) 2. \(x = y + z + 11 = 94 + 78 + 11 = 183\) 3. \(y = z + 16 = 78 + 16 = 94\) All equations are true, so our solution is correct.

Step 10: State the answer

Americans own a total of: - 183 million fish - 94 million cats - 78 million dogs

Key Concepts

Substitution Method

The substitution method is a powerful algebraic technique for solving systems of equations. It involves isolating one variable in terms of another, and then substituting that expression into the other equations. This conversion reduces the number of variables, simplifying the problem step by step.

In our exercise, we began by determining expressions for the number of cats and dogs so that we could replace those variables in the other equations. We took advantage of the given condition that 16 million more cats are owned than dogs, making it possible to write the equation for cats, namely, \(y = z + 16\).

With this equation for cats, we substituted it into the equation relating the fish to the number of cats and dogs, \(x = y + z + 11\). This substitution allowed us to express the number of fish as \(x = 2z + 27\), reducing our system of equations to two variables. With these simplifications, solving the equations becomes more straightforward.

Variables in Algebra

Variables are fundamental in algebra and serve as placeholders for unknown or changing values. They allow us to formulate equations to describe relationships between quantities. In this exercise, variables help represent the number of each type of pet: fish \(x\), cats \(y\), and dogs \(z\).

By defining these variables, we can create mathematical models that mirror real-world situations. This representation makes it easier to analyze and solve problems step by step.

Here's how we used each variable in this exercise:

• \(x\) (Fish): Represents the total number of fish owned, making it crucial to establish its relationship with \(y\) and \(z\) through equations.
• \(y\) (Cats): Helps to denote the number of cats, which in our problem is linked to \(z\), the number of dogs, with additional specifics provided.
• \(z\) (Dogs): The base variable we built upon to find both \(x\) and \(y\), as it is the simplest starting point because the condition \(y = z + 16\) was a straightforward expression.

Understanding how to define and utilize variables facilitates solving any algebraic system efficiently.

Algebraic Equations

Algebraic equations are equations wherein variables are related through mathematical operations. They are essential tools for problem-solving, transforming real-life situations into a mathematical framework that can be analyzed.

In our exercise, we formed three distinct algebraic equations based on the information given:

• The total number of pets \(x + y + z = 355\) helps aggregate the count of all pets, establishing the basis for solving other equations.
• The relation \(x = y + z + 11\) showed how the number of fish correlated with the sum of cats and dogs, incorporating the additional 11 million fish owned.
• Finally, \(y = z + 16\) defined the relationship between cats and dogs, which simplified the system by providing a direct equation to substitute when necessary.

These equations played key roles at different stages of solving the problem. They allowed us to express one variable in relation to others, simplify the system, and ultimately find the numerical values of each variable consistently. By understanding and using algebraic equations, complex systems can be addressed methodically.