Aufgaben zu linearen Gleichungssystemen

Aufgabe 1

a) \( a = 3 \), \( b = -1 \)

b) \( a = 5 \), \( b = 6 \)

c) \( a = \frac{1}{3} \), \( b = \frac{1}{3} \)

d) \( a = 5 \), \( b = 6 \)

e) \( a = 5 \), \( b = -2 \)

f) \( a = -7 \), \( b = 3 \)

g) \( a = 2 \), \( b = -1 \), \( c = 5 \)

Aufgabe 2

a) \( a = 5 \), \( b = 3 \)

b) \( a = -3 \), \( b = 1 \)

c) \( a = 3 \), \( b = -7 \)

d) \( a = 5 \), \( b = 6 \)

Aufgabe 3

a) \( a = -\frac{1}{4} \), \( b = -\frac{4}{3} \)

b) \( a = 5 \), \( b = 2 \)

c) \( a = 4 \), \( b = 0 \), \( c = 1 \)

d) \( a = -\frac{1}{3}b + \frac{2}{3}c \), \( b = b \), \( c = c \)
(Das Gleichungssystem hat unendlich viele Lösungen)

Aufgabe 4

a) \( a = \frac{29}{2} = 14.5 \), \( b = \frac{15}{2} = 7.5 \)

b) \( a = \frac{5}{4} - \frac{3}{4}b \), \( b = b \)
(Das Gleichungssystem hat unendlich viele Lösungen)

c) \( a = -\frac{2}{3} \), \( b = -\frac{7}{3} \)

d) \( a = \frac{2}{5} \), \( b = -\frac{3}{5} \)

e) \( a = 1 \), \( b = -1 \)

f) \( a = 3 \), \( b = 3 \)

Aufgabe 5

a) \( a = -1 \), \( b = 2.5 \), \( c = 1.5 \)

b) \( a = -2 \), \( b = 3 \), \( c = -1 \)

c) \( a = 0 \), \( b = -1 \), \( c = 2 \)

d) \( a = -1 \), \( b = -4 \), \( c = 3 \)

e) \( a = -1 \), \( b = 0 \), \( c = 1 \)

f) \( a = 1 \), \( b = -6 \), \( c = 9 \), \( d = 0 \)

g) \( a = -5 \), \( b = 4 \), \( c = 3 \)

h) keine Lösung (widersprüchliches Gleichungssystem)