# 9.6.3. Solving Two Equations in Two Variables, One Is of the First Degree, and the Other Is of the Second Degree

The length of a rectangle is cm, its width is cm, and its area is . If its length decreased by cm and its width increased by cm, it will become a square. Find the area of the square.

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### Example

The length of a rectangle is cm, its width is cm, and its area is . If its length decreased by cm and its width increased by cm, it will become a square. Find the area of the square.

### Solution

If the area of the rectangle is cm, then

The rectangle would become a square if its length were decreased by cm and its width increased by cm, then the two new dimensions would be the same. Hence,

This pair of equations can be solved by substitution. Rearranging the first order equation gives

Substituting into the second order equation gives

This equation can be solved by factorisation.

As represents the width of a rectangle, it must have a positive value. Therefore, cm.

The side length of the square is given by , and is therefore equal to cm.

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