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Mathematics
Problem
For a pair of linear equations
a
1
x
+
b
1
y
+
c
1
=
0
a_1x + b_1y + c_1 = 0
a
1
x
+
b
1
y
+
c
1
=
0
and
a
2
x
+
b
2
y
+
c
2
=
0
a_2x + b_2y + c_2 = 0
a
2
x
+
b
2
y
+
c
2
=
0
, which of the following conditions guarantees infinitely many solutions? a)
a
1
a
2
=
b
1
b
2
≠
c
1
c
2
\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}
a
2
a
1
=
b
2
b
1
=
c
2
c
1
b)
a
1
b
2
−
a
2
b
1
=
0
a_1b_2- a_2b_1=0
a
1
b
2
−
a
2
b
1
=
0
c)
a
1
a
2
=
b
1
b
2
=
c
1
c
2
\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}
a
2
a
1
=
b
2
b
1
=
c
2
c
1
d)
a
1
a
2
≠
b
1
b
2
\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}
a
2
a
1
=
b
2
b
1
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