Problem 198 - Elementary Algebra

Question

Riley is planning to plant a lawn in his yard. He will need nine pounds of grass seed. He wants to mix Bermuda seed that costs \(\$ 4.80\) per pound with Fescue seed that costs \(\$ 3.50\) per pound. How much of each seed should he buy so that the overall cost will be \(\$ 4.02\) per pound?

Step-by-Step Solution

Answer

3.60 pounds of Bermuda seed and 5.40 pounds of Fescue seed.

Step 1: Define Variables

Let \( x \) be the number of pounds of Bermuda seed, and \( y \) be the number of pounds of Fescue seed.

Step 2: Setup the Equation for Total Weight

Set up the equation for the total weight: \( x + y = 9 \). This represents the total weight of the seeds required.

Step 3: Setup the Equation for Total Cost

Set up the equation for the total cost: \( 4.80x + 3.50y = 4.02 \times 9 \). This represents the total cost of both types of seeds.

Step 4: Simplify the Total Cost Equation

Simplify the total cost equation: \( 4.80x + 3.50y = 36.18 \).

Step 5: Substitute and Solve for One Variable

Solve the weight equation for one variable: \( y = 9 - x \). Substitute \( y \) into the cost equation: \[ 4.80x + 3.50(9 - x) = 36.18 \].

Step 6: Simplify and Solve for x

Simplify the equation: \[ 4.80x + 31.50 - 3.50x = 36.18 \], \[ 1.30x + 31.50 = 36.18 \]. Solve for \( x \): \[ 1.30x = 4.68 \], \[ x = 3.60 \].

Step 7: Solve for y

Substitute \( x = 3.60 \) back into the weight equation: \( y = 9 - 3.60 \), \( y = 5.40 \).

Key Concepts

Linear Equations

A linear equation represents a straight-line relationship between two variables. In this problem, the total weight of the seeds forms a linear equation. The equation is written as \(x + y = 9\), where \(x\) is the pounds of Bermuda seed and \(y\) is the pounds of Fescue seed. This simple equation shows how the total weight stays constant when adding the two types of seeds together. Linear equations are fundamental in algebra, making it easier to solve problems involving relationships between quantities.

System of Equations

When you have more than one equation dealing with the same variables, it's called a system of equations. In this exercise, we need to consider both the total weight and total cost of the seeds. We have:

• The weight equation: \(x + y = 9\)
• The cost equation: \(4.80x + 3.50y = 36.18\)

This pair forms a system of linear equations. Solving such systems helps us find the exact quantities of each type of seed Riley should use. Systems of equations are crucial when dealing with multiple constraints or conditions.

Cost Analysis

Cost analysis involves calculating the total cost of items. Here, we calculate the total cost of seeds based on their prices per pound. Riley wants the overall cost to be \(\$4.02\) per pound. By setting the total cost equation \(4.80x + 3.50y = 36.18\), and solving it, we find the appropriate mix of Bermuda and Fescue seeds to achieve this. Performing cost analysis allows us to make informed decisions about purchases and optimize spending.

Substitution Method

The substitution method is a way to solve systems of equations. First, solve one equation for one variable. For example, \(y = 9 - x\) from the weight equation. Next, substitute this expression into the other equation:
\[4.80x + 3.50(9 - x) = 36.18\]
This gives you a single equation in one variable. Solve for that variable, then substitute back to find the other variable. This technique often simplifies complex systems, making them easier to solve step-by-step.