Find all integer pairs $(x, y)$ that satisfy $x + y = 12$ and $x y = 35$ .

hard 4 marks Class 7 · Mathematics · Integers

Answer

(7, 5)

and

(5, 7)

The correct answer is $(7, 5)$ and $(5, 7)$ — Thus the integer pairs are $(7, 5)$ and $(5, 7)$ .

Step-by-step solution

Treat the system as a quadratic: let $x$ be one variable; then $y = 12 - x$ .
Substitute into the product equation: $x (12 - x) = 35$ .
Rearrange to $x^{2} - 12 x + 35 = 0$ .
Factor the quadratic: $(x - 7) (x - 5) = 0$ .
Solve for $x$ : $x = 7$ or $x = 5$ .
Find corresponding $y$ using $y = 12 - x$ : if $x = 7$ , $y = 5$ ; if $x = 5$ , $y = 7$ .
Thus the integer pairs are $(7, 5)$ and $(5, 7)$ .

Common mistake: Missing negative roots or incorrect factorization

Concept tested: Solving simultaneous equations with integers