The Question
A rectangle having sides of (y – 3) metres and (x + 2) metres has a perimeter of 24 metres and an area of 32 square metres. Calculate the values of x and y.
Details
💡 Hint
Start by writing down the formulas for the perimeter and area of a rectangle. How can you use the given side expressions and total values to form two equations?
📝 Solution Steps
- Formulate the perimeter equation using the given side lengths and total perimeter.
- Formulate the area equation using the given side lengths and total area.
- From the perimeter equation, express one variable (e.g., y) in terms of the other (x).
- Substitute this expression into the area equation to get a single quadratic equation in one variable.
- Solve the quadratic equation for x.
- Substitute each value of x back into the linear equation to find the corresponding y values.
📚 Explanation
This problem requires setting up and solving a system of simultaneous equations. First, translate the given information about the rectangle’s perimeter and area into two algebraic equations. The perimeter of a rectangle is 2(length + width), and the area is length × width. Let the sides be (y – 3) and (x + 2). This gives you a linear equation for the perimeter and a quadratic equation for the area. From the linear (perimeter) equation, express one variable in terms of the other (e.g., y in terms of x). Substitute this expression into the quadratic (area) equation. Solve the resulting quadratic equation for x. You might get two possible values for x. For each value of x, substitute it back into the linear equation to find the corresponding value of y. Remember to check if both solutions make sense in the context of side lengths (i.e., side lengths must be positive).
✅ Answer
x = 6, y = 7 or x = 2, y = 11
⚠️ Common Mistakes
- Incorrectly setting up the perimeter or area equations.
- Algebraic errors during substitution or when solving the quadratic equation.
- Forgetting to find both x and y values for each valid solution.
- Not checking if the calculated side lengths are positive.
📐 Formulas Required
- Perimeter of rectangle = 2(length + width)
- Area of rectangle = length × width
