Question

Solve the following simultaneous equations using Cramer's rule .
(v) x + 18 = 2 y ; y = 2x $-$ 9

Answer 1

The given equations are:

x + 18 = 2 y
⇒ x $-$ 2 y = $-$ 18 ----------(i)
and
y = 2x $-$ 9
⇒2x $-$ y = 9 ----------(ii)

Here, D = $\left|\begin{array}{cc}1& -2\\ 2& -1\end{array}\right|=1\times \left(-1\right)-\left(-2\right)\times 2=-1+4=3$

Now, we have:
D_x =$\left|\begin{array}{cc}-18& -2\\ 9& -1\end{array}\right|=\left(-18\right)\times \left(-1\right)-\left(-2\right)\times 9=18+18=36$

D_y =$\left|\begin{array}{cc}1& -18\\ 2& 9\end{array}\right|=1\times 9-\left(-18\right)\times 2=9+36=45$

By Cramer's rule, we get:

$$\begin{gathered} x=\frac{D_x}{D}=\frac{36}{3}=12 \\ \mathrm{and} \\ y=\frac{Dy}{D}=\frac{45}{3}=15 \\ \therefore x=12\mathrm{and}y=15\mathrm{is}\mathrm{the}\mathrm{solution}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{simultaneous}\mathrm{equations}. \end{gathered}$$