Question

The pair of equations $5x-15y=8$ and $3x-9y=\frac{24}{5}$ has,

• A. One solution
• B. Two solutions
• C. Infinitely many solutions
• D. no solution

Answer 1

The correct option is C

Infinitely many solutions

The explanation for the correct option.

If the equation of two lines are,

$$\begin{gathered} a_{1}x+b_{1}y+c_1=0 \\ {a}_{2}x+b_{2}y+c_2=0 \end{gathered}$$

• If $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$, the two lines are parallel and have no solutions.
• If $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$, the two lines are coincident and have infinite solutions.
• If $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$, the two lines are intersecting and have unique solutions.

The pair of equations are $5x-15y=8$ and $3x-9y=\frac{24}{5}$.

Compared with the standard form,

$$\begin{gathered} a_1=5,b_1=-15,c_1=8 \\ {a}_2=3,b_2=-9,c_2=\frac{24}{5} \end{gathered}$$

$$\begin{gathered} \frac{a_1}{a_2}=\frac{5}{3} \\ \frac{b_1}{b_2}=\frac{-15}{-9}=\frac{5}{3} \\ \frac{c_1}{c_2}=\frac{8}{24/5}=\frac{5}{3} \end{gathered}$$

Thus,$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$.

Therefore, the given pair of equations has infinite solutions.

Hence, the option $\left(C\right)$ is correct.