Question

Solve for the unknown variables:
(a) (a, b) + (3, 1) = (6, 7)
(b) (11, c) + (d, 6) = (-12, −3)
(c) (5, 5) + (e, 4) = (2, f)
(d) (8, −3) − (5, g) = (h, 2)

Answer 1

1. $(a,b) + (3,1) = (6,7)$

$$(a, b) = (6,7) - (3,1)$$ $$(a, b) = (3,6)$$

Answer: $a = 3$, $b = 6$

2. $(11, c) + (d, 6) = (-12, -3)$

$$(11 + d, c + 6) = (-12, -3)$$ $$11 + d = -12 \quad \Rightarrow \quad d = -23$$ $$c + 6 = -3 \quad \Rightarrow \quad c = -9$$

Answer: $c = -9$, $d = -23$

3. $(5,5) + (e,4) = (2,f)$

$$(5,5) = (2, f) - (e, 4)$$ $$(5,5) = (2 - e, f - 4)$$ $$5 = 2 - e \quad \Rightarrow \quad e = -3$$ $$5 = f - 4 \quad \Rightarrow \quad f = 9$$

Answer: $e = -3$, $f = 9$

4. $(8, -3) - (5, g) = (h, 2)$

$$(3, -3 - g) = (h, 2)$$ $$3 = h$$ $$-3 - g = 2 \quad \Rightarrow \quad g = -5$$

Answer: $h = 3$, $g = -5$