Problem 122 - College Algebra with Modeling and Visualization

Question

During the first four months of \(2004,\) a total of 6767 people were vaccinated for smallpox in Florida and Texas. There were 273 more people vaccinated in Florida than in Texas. (Source: CDC.) (a) Write a system of equations whose solution represents the number of vaccinations given in each state. (b) Solve the system symbolically. (c) Solve the system graphically.

Step-by-Step Solution

Answer

Texas: 3247 vaccinations; Florida: 3520 vaccinations.

Step 1: Understanding the Variables

Let's assign variables for the number of vaccinations in each state. Let \( F \) represent the number of people vaccinated in Florida, and \( T \) represent the number of people vaccinated in Texas.

Step 2: Formulating the System of Equations

From the problem, we know two things: the total number of vaccinations in both states is 6767, and Florida had 273 more vaccinations than Texas. This gives us the equations: \( F + T = 6767 \) and \( F = T + 273 \).

Step 3: Solving the System Symbolically - Substitution Method

We can solve the equations by substitution. Substitute \( F = T + 273 \) into \( F + T = 6767 \), resulting in:\[ (T + 273) + T = 6767 \]Simplify and solve for \( T \):\[ 2T + 273 = 6767 \]Subtract 273 from both sides:\[ 2T = 6494 \]Finally, divide by 2:\[ T = 3247 \]

Step 4: Finding the Number of Vaccinations in Florida

Now that we have \( T = 3247 \), substitute this back into \( F = T + 273 \) to find \( F \):\[ F = 3247 + 273 = 3520 \]

Step 5: Verifying the Solution

Verify the solution by checking if the total adds up to 6767:\[ F + T = 3520 + 3247 = 6767 \]Both equations and conditions are satisfied.

Step 6: Solving the System Graphically

To solve graphically, plot the equations \( F = 6767 - T \) and \( F = T + 273 \) on the same graph, where the x-axis is \( T \) and the y-axis is \( F \). The point where they intersect gives the solution \( (T, F) = (3247, 3520) \).

Key Concepts

Solving Algebraically

Solving systems of equations algebraically involves using mathematical techniques to find the values of variables that satisfy all equations simultaneously. In the context of our exercise, we're given two equations:

• \( F + T = 6767 \)
• \( F = T + 273 \)

To solve these algebraically, we often use methods like substitution or elimination. Here, the substitution method is applied:

• First, express one variable in terms of the other from one equation.
• Next, substitute this expression into the second equation to find one variable's value.
• Finally, substitute this value back to find the other variable.

This method provides a precise numerical solution, bypassing the need for visual tools like graphs. In our problem, we successfully find that \( T = 3247 \) and \( F = 3520 \). These figures represent the vaccinations in Texas and Florida, respectively.

Solving Graphically

Graphical solutions offer a visual representation of systems of equations by plotting each equation on a coordinate plane. For many students, this can make abstract concepts more concrete and intuitive. In the exercise:

• Draw lines for the equations \( F = 6767 - T \) and \( F = T + 273 \).
• Each line represents all the possible solutions to one equation.
• The intersection point where these two lines meet is the solution that satisfies both equations simultaneously.

This method is particularly helpful when dealing with equations that might be complex algebraically. However, it's essential to use an accurate scale to ensure that where the lines intersect is clear. Graphical solutions are a great way to double-check the results achieved using algebraic methods.

Word Problems

Word problems translate real-life scenarios into mathematical equations. It involves identifying key quantities and relationships described in the problem and expressing them with variables and equations. When tackling such problems:

• Read the problem carefully to understand what is being asked.
• Identify and define variables to represent unknowns.
• Formulate equations based on relationships and constraints mentioned in the problem.
• Solve the equations using suitable methods.

In our exercise, the key step was recognizing the total vaccinations and the difference in vaccinations between the two states, then converting those into a system of equations. Mastery in this area often requires practice in interpreting context and translating it into a mathematical framework.

Substitution Method

The substitution method is a straightforward technique used to solve systems of linear equations. It involves solving one equation for one variable and substituting this expression into the other equation. Here's how it works:

• First, isolate one variable in either equation.
• Use this expression to replace the same variable in the other equation.
• Solve this new equation for the remaining variable.
• Finally, substitute back to find the value of the first variable.

In our example, once we write \( F = T + 273 \), substituting this into \( F + T = 6767 \) simplifies the system to a single equation in \( T \). This method is beneficial when equations are easily solvable or when they are already in a manageable form for substitution. It requires careful algebraic manipulation to ensure accuracy.