Question

If a system of linear equations consists of two lines that are exactly the same (overlap; produce the same graph), how many solutions does the system have?
0

1

2

infinite

Answer 1

Consider the system of linear equation:

$\left\{\begin{array}{l}a_1x+b_1y=c_1\\a_2x+b_2y=c_2\end{array}\right.$

This system will have:

• exactly one solution if $\dfrac{a_1}{a_2}\neq \dfrac{b_1}{b_2};$
• no solutions if $\dfrac{a_1}{a_2}= \dfrac{b_1}{b_2}\neq \dfrac{c_1}{c_2};$
• infinetely many solutions if $\dfrac{a_1}{a_2}= \dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}.$

If a system of linear equations consists of two lines that are exactly the same ($a_1x+b_1y=c_1$), then $\dfrac{a_1}{a_1}= \dfrac{b_1}{b_1}=\dfrac{c_1}{c_1}=1$ and  the system will have infinitely many solutions.

Answer: correct choice is D.