alleryA system of two linear equations in two variables has infinitely many solutions when which condition is satisfied?
A$\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$
B$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}$
C$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
D$a_1a_2+b_1b_2=c_1c_2$
Answer & Solution
Correct answer: C. $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
A pair of linear equations has infinitely many solutions when the two equations represent the same line.
Algebraically, this happens when the ratios of corresponding coefficients and constants are all equal: $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$.
**Option B instead gives no solution, because the lines are parallel but distinct.**
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