Problem 87 - Elementary Algebra

Question

A pickup truck with \(45 \mathrm{ft}^{3}\) of compost in its bed can be emptied with 3 loads of a garden cart and 10 loads of a wheelbarrow, or it can be emptied with 6 loads of a garden cart and 5 loads of a wheelbarrow. Use a system of two linear equations to find the volume of compost carried by the garden cart and by the wheelbarrow.

Step-by-Step Solution

Answer

G (garden cart) = 5 W (wheelbarrow) = 3

Step 1 - Define Variables

Let the volume of compost carried by the garden cart be denoted as G (in cubic feet), and the volume of compost carried by the wheelbarrow be denoted as W (in cubic feet).

Step 2 - Set Up Equations

According to the problem, the pickup truck can be emptied with either 3 loads of a garden cart and 10 loads of a wheelbarrow or 6 loads of a garden cart and 5 loads of a wheelbarrow. Therefore, we can set up the following system of linear equations: 1) 3G + 10W = 452) 6G + 5W = 45

Step 3 - Multiply Equation 1

To eliminate one variable, multiply the first equation by 2. This gives:6G + 20W = 90

Step 4 - Subtract Equations

Subtract the second equation from the modified first equation to eliminate G:(6G + 20W) - (6G + 5W) = 90 - 4515W = 45

Step 5 - Solve for W

Solve for W by dividing both sides of the equation by 15:W = 3

Step 6 - Substitute W Back into One of the Original Equations

Substitute W = 3 into the first equation to find G:3G + 10(3) = 453G + 30 = 453G = 15G = 5

Key Concepts

algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In this exercise, algebra helps us find unknown values through equations.

The problem gives us two scenarios involving a pickup truck, a garden cart, and a wheelbarrow to transport compost. By representing the unknown values (volumes of the garden cart and the wheelbarrow) with variables, we can create equations to solve for these unknowns.

Using algebra, we define the volumes of compost carried by the garden cart as G and by the wheelbarrow as W. This ability to set up and solve equations is a fundamental skill in algebra.

linear equations

Linear equations are equations of the first order, meaning each term is either a constant or the product of a constant and a single variable. The system of linear equations in this exercise is used to solve real-world problems, like determining the volume of compost carried by the garden cart and the wheelbarrow.

We are given:
1) 3G + 10W = 45
2) 6G + 5W = 45

These equations are linear because they represent straight lines. To solve for G and W, we can use methods such as substitution or elimination. In this case, we used elimination by first multiplying the first equation by 2, giving us 6G + 20W = 90, then subtracting the second equation from this modified equation to isolate one of the variables.

The final step involves simple arithmetic to find the values of W and G.

problem-solving

Problem-solving in mathematics often involves a systematic approach to break down complex problems into manageable steps. In this exercise, we use a step-by-step method to solve a system of linear equations.

Here’s how it works:

• Step 1: Define the variables (G for the garden cart and W for the wheelbarrow).
• Step 2: Set up the equations based on given information.
• Step 3: Use elimination or substitution to isolate one variable.
• Step 4: Substitute back to find the other variable.

This process ensures that each part of the problem is tackled in an organized manner. By carefully following these steps, we can find accurate solutions and understand how different parts of the problem relate.

This structured approach not only helps in solving equations but also improves analytical thinking skills applicable in various fields.