Question

A system of two linear equations
$y+3x=-3$
$2y=cx+d$
has no solutions. Then, find the value of $c$ and $d$

• A. $c\ne -6$, $d\ne -6$
• B. $c=-6$, $d\ne -6$
• C. $c\ne -6$, $d=-6$
• D. $c=-6$, $d=-6$

Answer 1

The correct option is B $c=-6$, $d\ne -6$

Given equations :
$y+3x=-3\dots \left(i\right)$
$2y=cx+d\dots \left(ii\right)$
Writing above eqations in the form of $y=mx+b$.
So,
Equation $\left(i\right)$ $\to y=-3x-3$
Equation $\left(ii\right)$ $\to y=\frac{c}{2}x+\frac{d}{2}$
In case of no solution, their slopes must be equal and their intercepts must be different
Equating slopes of two equations,
$-3=\frac{c}{2}$
$c=-3\times 2$
$c=-6$

And their intercept should not be the same
Hence,
$-3\ne \frac{d}{2}$
$d\ne -3\times 2$
$d\ne -6$

Answer: $c=-6$ and $d\ne -6$