Problem 53 - Introductory Algebra for College Students

Question

Tourist: "How many birds and lions do you have in your zoo?" Zookeeper: "There are 30 heads and 100 feet." Tourist: "I can't tell from that." Zookeeper: "Oh, yes, you can!" Can you? Find the number of each.

Step-by-Step Solution

Answer

The zoo has 10 birds and 20 lions.

Step 1: Formulate The Equations

From the problem, we can establish two equations. Let's represent the number of birds as \(x\) and the number of lions as \(y\). Equation 1: \(x+y=30\) (Since the total number of heads (birds + lions) is equal to 30). Equation 2: \(2x+4y=100\) (since the total number of feet (2 feet/ bird + 4 feet/lion) is equal to 100)

Step 2: Simplify The Equations

The second equation can be simplified: both sides of \(2x+4y=100\) can be divided by 2 to make it simpler. Therefore we get a new form of the second equation: \(x+2y=50\).

Step 3: Solve Using Substitution or Elimination

Now we have two equations: \(x+y=30\) and \(x+2y=50\). We can solve these using the substitution or elimination method. In this case, we will use the elimination method, we subtract the first equation from the new second equation to get: \(x+2y-x-y=50-30\) leading to \(y=20\). Substitute \(y=20\) into Equation 1, we get: \(x+20=30\), which leads to \(x=10\).

Key Concepts

Algebraic Equations

At the heart of many mathematical problems – particularly those involving scenarios with constraints – are algebraic equations. An algebraic equation is a statement of equality between two expressions that contain variables and constants.

For instance, if a zookeeper tells you there are 30 heads and 100 feet among the birds and lions in the zoo, you can turn that information into algebraic equations. Let's denote the number of birds as x and the number of lions as y. The 'heads' information becomes the first equation, x + y = 30, where each animal, regardless of being a bird or lion, contributes one head to the count. The 'feet' information translates to 2x + 4y = 100, as birds have 2 feet and lions have 4.

These equations provide a structured way to describe the situation, setting up the basis for finding a solution using algebraic methods such as substitution or elimination.

Elimination Method

The elimination method is one of the algebraic tools we can use to solve systems of linear equations. In the case of our zoo scenario with birds and lions, this method involves manipulating the equations to eliminate one of the variables, allowing us to find the value of the other.

To apply the elimination method, we often manipulate the equations to have opposite coefficients in front of one of the variables. In our example, we have x + y = 30 and x + 2y = 50 after simplifying the original 'feet' equation by dividing everything by 2. To eliminate x, we can subtract the first equation from the second, resulting in y = 20. This is a clear and straightforward approach when the equations are set up favorably, as they are in this problem. Subtracting removes x entirely and gives us the number of lions directly.

The elimination method is particularly helpful when the system of equations is simple to manipulate, often being faster than substitution when dealing with larger systems or when the coefficients are easily aligned.

Substitution Method

Another powerful technique in solving systems of equations is the substitution method. This method involves solving one of the equations for one variable and then substituting that expression into the other equation.

In the context of our zoo example, if we had chosen substitution over elimination, we might solve the first equation for x, getting x = 30 - y. Then we would replace x in the second equation with 30 - y, leading to 2(30 - y) + 4y = 100, which could then be simplified to find the value for y.

The substitution method is especially useful when one of the equations is already solved for a variable or can be manipulated with relative ease to be in that form. It is a method that can provide a clear path to finding one variable at a time, which can be less intimidating for those who are new to algebra.