Question

Solve the following simultaneous equations using Cramer’s rule.
(1) 3x – 4y = 10 ; 4x + 3y = 5
(2) 4x + 3y – 4 = 0 ; 6x = 8 – 5y
(3) x + 2y = –1 ; 2x – 3y = 12
(4) 6x – 4y = –12 ; 8x – 3y = –2
(5) 4m + 6n = 54 ; 3m + 2n = 28
(6) $2x+3y=2;x-\frac{y}{2}=\frac{1}{2}$

Answer 1

(1) 3x – 4y = 10
4x + 3y = 5
$$\begin{gathered} D=\left|\begin{array}{cc}3& -4\\ 4& 3\end{array}\right|=3\times 3-\left(-4\right)\times 4=9+16=25 \\ {D}_x=\left|\begin{array}{cc}10& -4\\ 5& 3\end{array}\right|=10\times 3-\left(-4\right)\times 5=30+20=50 \\ {D}_y=\left|\begin{array}{cc}3& 10\\ 4& 5\end{array}\right|=3\times 5-10\times 4=15-40=-25 \end{gathered}$$
$$\begin{gathered} x=\frac{D_x}{D}=\frac{50}{25}=2 \\ y=\frac{D_y}{D}=\frac{-25}{25}=-1 \\ \left(x,y\right)=\left(2,-1\right) \end{gathered}$$

(2) 4x + 3y – 4 = 0 ; 6x = 8 – 5y
$$\begin{gathered} D=\left|\begin{array}{cc}4& 3\\ 6& 5\end{array}\right|=4\times 5-6\times 3=20-18=2 \\ {D}_x=\left|\begin{array}{cc}4& 3\\ 8& 5\end{array}\right|=4\times 5-3\times 8=20-24=-4 \\ {D}_y=\left|\begin{array}{cc}4& 4\\ 6& 8\end{array}\right|=4\times 8-6\times 4=32-24=8 \end{gathered}$$
$$\begin{gathered} x=\frac{D_x}{D}=\frac{-4}{2}=-2 \\ y=\frac{D_y}{D}=\frac{8}{2}=4 \\ \left(x,y\right)=\left(-2,4\right) \end{gathered}$$

(3) x + 2y = –1 ; 2x – 3y = 12
$$\begin{gathered} D=\left|\begin{array}{cc}1& 2\\ 2& -3\end{array}\right|=1\times \left(-3\right)-2\times 2=-3-4=-7 \\ {D}_x=\left|\begin{array}{cc}-1& 2\\ 12& -3\end{array}\right|=\left(-1\right)\times \left(-3\right)-2\times 12=3-24=-21 \\ {D}_y=\left|\begin{array}{cc}1& -1\\ 2& 12\end{array}\right|=1\times 12-\left(-1\right)\times 2=12+2=14 \end{gathered}$$
$$\begin{gathered} x=\frac{D_x}{D}=\frac{-21}{-7}=3 \\ y=\frac{D_y}{D}=\frac{14}{-7}=-2 \\ \left(x,y\right)=\left(3,-2\right) \end{gathered}$$

(4) 6x – 4y = –12 ; 8x – 3y = –2

$$\begin{gathered} D=\left|\begin{array}{cc}6& -4\\ 8& -3\end{array}\right|=6\times \left(-3\right)-\left(-4\right)\times 8=-18+32=14 \\ {D}_x=\left|\begin{array}{cc}-12& -4\\ -2& -3\end{array}\right|=\left(-12\right)\times \left(-3\right)-\left(-4\right)\times \left(-2\right)=36-8=28 \\ {D}_y=\left|\begin{array}{cc}6& -12\\ 8& -2\end{array}\right|=6\times \left(-2\right)-\left(-12\right)\times 8=-12+96=84 \end{gathered}$$
$$\begin{gathered} x=\frac{D_x}{D}=\frac{28}{14}=2 \\ y=\frac{D_y}{D}=\frac{84}{14}=6 \\ \left(x,y\right)=\left(2,6\right) \end{gathered}$$

(5) 4m + 6n = 54 ; 3m + 2n = 28
$$\begin{gathered} D=\left|\begin{array}{cc}4& 6\\ 3& 2\end{array}\right|=4\times 2-6\times 3=8-18=-10 \\ {D}_x=\left|\begin{array}{cc}54& 6\\ 28& 2\end{array}\right|=54\times 2-6\times 28=108-168=-60 \\ {D}_y=\left|\begin{array}{cc}4& 54\\ 3& 28\end{array}\right|=4\times 28-54\times 3=112-162=-50 \end{gathered}$$
$$\begin{gathered} x=\frac{D_x}{D}=\frac{-60}{-10}=6 \\ y=\frac{D_y}{D}=\frac{-50}{-10}=5 \\ \left(x,y\right)=\left(6,5\right) \end{gathered}$$

(6) $2x+3y=2;x-\frac{y}{2}=\frac{1}{2}$
$$\begin{gathered} D=\left|\begin{array}{cc}2& 3\\ 1& \frac{-1}{2}\end{array}\right|=2\times \left(-\frac{1}{2}\right)-3\times 1=-1-3=-4 \\ {D}_x=\left|\begin{array}{cc}2& 3\\ \frac{1}{2}& \frac{-1}{2}\end{array}\right|=2\times \left(-\frac{1}{2}\right)-3\times \frac{1}{2}=-1-\frac{3}{2}=\frac{-5}{2} \\ {D}_y=\left|\begin{array}{cc}2& 2\\ 1& \frac{1}{2}\end{array}\right|=2\times \frac{1}{2}-2\times 1=1-2=-1 \end{gathered}$$
$$\begin{gathered} x=\frac{D_x}{D}=\frac{{\displaystyle \frac{-5}{2}}}{-4}=\frac{5}{8} \\ y=\frac{D_y}{D}=\frac{-1}{-4}=\frac{1}{4} \\ \left(x,y\right)=\left(\frac{5}{8},\frac{1}{4}\right) \end{gathered}$$