Problem 13 - Algebra A Combined Function

Question

In 2008 , New Mexico produced 15 million pounds more pecans than Texas. Together, the two states produced 75 million pounds of pecans. Find the amount of pecans grown in New Mexico and Texas in 2008\. (Source: National Agriculture Statistics Service)

Step-by-Step Solution

Answer

Texas produced 30 million pounds and New Mexico produced 45 million pounds.

Step 1: Define Variables

Let's define the variables to represent the unknown quantities. Let \( x \) be the amount of pecans (in million pounds) produced by Texas in 2008. Then the amount produced by New Mexico would be \( x + 15 \) million pounds.

Step 2: Set Up the Equation

According to the problem, the total production of pecans by both states together is 75 million pounds. We can express this as an equation: \( x + (x + 15) = 75 \). Simplifying, we get \( 2x + 15 = 75 \).

Step 3: Solve the Equation

Solve the equation \( 2x + 15 = 75 \) to find \( x \). First, subtract 15 from both sides to get \( 2x = 60 \). Then divide both sides by 2 to get \( x = 30 \).

Step 4: Find the Pecans Produced by New Mexico

Since \( x = 30 \), New Mexico produced \( x + 15 = 30 + 15 = 45 \) million pounds of pecans.

Key Concepts

Linear Equations

Linear equations are simple equations that involve a single variable and have a degree of 1. They are fundamental to solving word problems in algebra. In the exercise given, the main linear equation we derive is:

• \( x + (x + 15) = 75 \)

This equation represents the total amount of pecans produced by both states. The term \( x \) represents Texas's production, and \( x + 15 \) represents New Mexico's production. Linear equations are used to model real-life situations like this to find unknown quantities.

Variable Definition

Defining variables is the first step in solving algebra word problems. By clearly defining what our variables stand for, we make it easier to set up and solve equations.

• In our example, we defined \( x \) as the amount of pecans produced by Texas.
• New Mexico's production is defined in terms of \( x \) as \( x + 15 \).

Defining variables correctly helps translate verbal problems into mathematical expressions. Always ensure that each variable uniquely represents one specific quantity in the problem.

Problem-solving Steps

Solving a word problem involves a series of organized steps that lead to the solution.

• Step 1: Read the problem carefully and identify what is being asked.
• Step 2: Define the variables based on the given conditions.
• Step 3: Set up an equation or equations using these variables.
• Step 4: Simplify and solve the equation(s) to find the values of the variables.
These steps provide a structured approach to solving even the most complex algebraic word problems. Following these steps ensures nothing is missed in the problem-solving process.

Equation Simplification

Simplifying equations is a crucial step in solving any mathematical problem. It involves reducing the complexity of the equation to make it easier to solve.

• In the given example, the initial equation \( x + (x + 15) = 75 \) simplifies to \( 2x + 15 = 75 \).
• We then subtract 15 from both sides to get \( 2x = 60 \).
• Finally, we divide both sides by 2 to find \( x = 30 \).

These simplification steps make solving equations straightforward, helping to find exact solutions quickly. Simplification helps manage the problem complexity by making equations more manageable.