Question

If $a_1{b}_2\ne a_2{b}_1$, then the system of equations $a_{1}x+b_{1}y=c_1$ and $a_{2}x+b_{2}y=c_2$

• A. has a unique solution
• B. has no solution
• C. has infinitely many solutions
• D. has two solutions

Answer 1

The correct option is A has a unique solution

A system of linear equations $ax+by+c=0$ and $dx+ey+g=0$ will have a unique solution if the two lines represented by the equations $ax+by+c=0$ and $dx+ey+g=0$ intersect at a point i.e., if the two lines are neither parallel nor coincident.

Essentially, the slopes of the two lines should be different.

Writing the equations in slope form we get

$y=-\frac{a_{1}x}{b_1}+\frac{c_1}{b_1}$

$y=-\frac{a_{2}x}{b_2}+\frac{c_2}{b_2}$

For a unique solution, the slopes of the lines should be different.

$-\frac{a_1}{b_1}\ne -\frac{a_2}{b_2}$

$\frac{a_1}{b_1}\ne \frac{a_2}{b_2}$

Hence $a_1{b}_2\ne a_2{b}_1$