Garden Dimensions and Algebraic Problem Solving: A Step-by-Step Guide

When dealing with algebraic problems, especially those involving the dimensions of a garden, a clear understanding of the steps to solve the equation is crucial. This article explores a common scenario where the length of a garden is 5 meters longer than its width, and the area is given. The process involves setting up the equation, solving it, and validating the solution through check-ups. Let's delve into the process.

Solution 1: Using Quadratic Equations

Let the width of the garden be w meters. Then the length of the garden can be expressed as l w 5 meters. The area of the garden is given by the formula: Area length * width. Substituting the expressions for length and area, we get:

6 w 5 * w Pose the equation: 6 w^2 5w Rearrange the equation to: w^2 5w - 6 0 Factor the quadratic equation: (w 6)(w - 1) 0 Solve for w using the zero product rule: w 6 0, which gives w -6, not a valid solution since the width cannot be negative. w - 1 0, which gives w 1.

With w 1, we can find the length:

l w 5 1 5 6 meters. The length of the garden is 6 meters.

This solution is validated by checking the area: 1 * 6 6, which matches the given area.

Solution 2: Alternative Quadratic Equation Method

Let the width of the garden be w meters. Then the length of the garden can also be expressed as l w 5 meters. Given that the area is 6 square meters, we can set up the equation as follows:

w * (w 5) 6 Expand the equation: w^2 5w 6 Rearrange the equation to: w^2 5w - 6 0 Solve the quadratic equation by factoring: (w 6)(w - 1) 0 Find the solutions for w using the zero product rule: w 6 0, which gives w -6, not a valid solution since the width cannot be negative. w - 1 0, which gives w 1.

With w 1, we can find the length:

l w 5 1 5 6 meters. The length of the garden is 6 meters.

Solution 3: Another Approach

Let the width of the garden be x meters. Then the length of the garden can be expressed as l x 5 meters. Given that the area is 6 square meters, we can set up the equation as follows:

x * (x 5) 36 Expand the equation: x^2 5x 36 Rearrange the equation to: x^2 5x - 36 0 Factor the quadratic equation: (x 9)(x - 4) 0 Find the solutions for x using the zero product rule: x 9 0, which gives x -9, not a valid solution since the width cannot be negative. x - 4 0, which gives x 4.

With x 4, we can find the length:

l x 5 4 5 9 meters.

The length of the garden is 9 meters.

Conclusion

These examples demonstrate the importance of a well-structured approach to solving algebraic problems. Whether using the width as w or x, the steps and the logic remain consistent. By understanding the fundamental principles of algebra, we can accurately solve for the dimensions of a garden and similar geometric problems. If you have a similar problem or need help with other algebraic equations, the techniques demonstrated here can guide you through the solution process.