Problem 13 - Intermediate Algebra

Question

Jerry Rice, who played the majority of his career with the San Francisco 49 ers and the Oakland Raiders, holds the all-time record for touchdown (TD) passes caught. Here are some interesting facts about this feat. -He caught 30 more TD passes from Steve Young than he did from Joe Montana. -He caught 39 more TD passes from Joe Montana than he did from Rich Gannon. -He caught a total of 156 TD passes from Young, Montana, and Gannon. Determine the number of touchdown passes Rice has caught from Young, from Montana, and from Gannon.

Step-by-Step Solution

Answer

Rice caught 85 touchdowns from Young, 55 from Montana, and 16 from Gannon.

Step 1: Define Variables

Let's define three variables: \( Y \) for the number of touchdowns caught from Steve Young, \( M \) for the touchdowns from Joe Montana, and \( G \) for the touchdowns from Rich Gannon.

Step 2: Set Up Equations from Given Information

1. From the problem, we know Rice caught 30 more touchdowns from Young than Montana. So, we write the equation: \( Y = M + 30 \).2. He caught 39 more touchdowns from Montana than from Gannon: \( M = G + 39 \).3. The total number of touchdowns caught from these three quarterbacks is 156: \( Y + M + G = 156 \).

Step 3: Substitute and Solve for Variables

Substitute \( Y = M + 30 \) and \( M = G + 39 \) into the total touchdowns equation.\((M + 30) + M + G = 156\)\Substitute \( M = G + 39 \) into the equation:\((G + 39 + 30) + (G + 39) + G = 156\)\Simplify the equation:\(3G + 108 = 156\)\Solve for \( G \):\(3G = 156 - 108\)\(3G = 48\)\(G = 16\)

Step 4: Substitute Back to Find M and Y

Now that we have \( G = 16 \), substitute it back to find \( M \):\( M = G + 39 = 16 + 39 = 55 \).\Next, substitute \( M \) to find \( Y \):\( Y = M + 30 = 55 + 30 = 85 \).

Step 5: Verify the Solution

Verify that the numbers add up properly:1. \( Y = 85 \), \( M = 55 \), \( G = 16 \)2. Total touchdowns: \( 85 + 55 + 16 = 156 \)3. Verify conditions: - \( Y = M + 30 \), which is \( 85 = 55 + 30 \), correct. - \( M = G + 39 \), which is \( 55 = 16 + 39 \), correct.All conditions are met.

Key Concepts

Understanding the Substitution Method

The substitution method is a powerful tool used to solve systems of linear equations. It's like solving a puzzle by replacing a piece you know with something else that fits. Imagine you have several equations that need to be solved for multiple variables, which means there are multiple unknowns like the variables in a math problem.

• The main idea is to solve one of the equations for one variable.
• This solution is then substituted into the other equations.
• The process repeats until you have a value for each variable you're looking for.

This method works well when one of the equations is easy to solve for one variable. It helps in reducing the number of unknowns step by step, making the problem more manageable. For instance, if you know that \( Y = M + 30 \), you can eliminate \( Y \) from other equations by replacing it with \( M + 30 \). This way, you simplify to fewer variables, making the math easier.
The substitution method is systematic and logical, ensuring that if followed correctly, the solution will satisfy all equations in the system.

The Art of Algebraic Problem Solving

Algebraic problem-solving involves finding solutions to sets of equations or mathematical statements that have unknown variables. It's the backbone of algebra and develops logical thinking and reasoning.
When dealing with algebra, especially systems of equations, here are key steps you should follow:

• First, clearly define all variables. Without clear definitions, solving becomes confusing.
• Next, translate word problems into mathematical equations. Every part of the problem should have a corresponding mathematical expression.
• Then, use methods like substitution or elimination to simplify and solve equations.
• Finally, always verify your solutions to check that they satisfy the original equations.

In our example, defining variables as touchdowns from Young, Montana, and Gannon allowed us to construct meaningful equations. Setting up the relationships between these touchdowns using subtraction and addition based on given specific number differences turns abstract problems into manageable math expressions.
Algebraic problem-solving is not just about numbers; it's about connections and relationships among those numbers to uncover values of unknowns.

Introduction to Linear Equations

Linear equations form the foundation of many algebraic solutions. They are equations where each term is either a constant or the product of a constant and a single variable.
Some features of linear equations include:

• They graph as straight lines in the coordinate plane.
• Each solution to a linear equation gives a point on this line.
• Equations are often expressed in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.

In the context of the touchdown problem, the equations we formed, such as \( Y = M + 30 \), \( M = G + 39 \), and \( Y + M + G = 156 \), are all linear. When these equations are solved together, they lead to intersections that provide the solutions to the touchdowns caught from each quarterback.
Linear equations can be solved using several algebraic methods, from graphing to algebraic manipulation like substitution.
Understanding how to form and solve these equations is crucial for tackling various real-world problems modeled by algebra. They are essential because many natural and economic phenomena can be modeled using linear models.