Question

Solve the following sets of simultaneous equations.

(i) x + y = 4 ; 2x $-$ 5y = 1

(ii) 2x + y = 5; 3x $-$ y = 5

(iii) 3x$-$ 5y =16; x $-$ 3y = 8

(iv) 2y$-$x =0; 10x + 15y = 105

(v) 2x + 3y + 4 = 0; x$-$ 5y = 11

(vi) 2x $-$7y = 7; 3x + y = 22

Answer 1

(i) x + y = 4 .....(I)
2x $-$ 5y = 1 .....(II)
From (I) we have
$x=4-y$
Putting this value of x in (II)
$$\begin{gathered} 2\left(4-y\right)-5y=1 \\ \Rightarrow 8-2y-5y=1 \\ \Rightarrow 8-7y=1 \\ \Rightarrow 8-1=7y \\ \Rightarrow 7=7y \\ \Rightarrow y=1 \end{gathered}$$
Putting the value of y in (I) we have
$$\begin{gathered} x+1=4 \\ \Rightarrow x=3 \end{gathered}$$
Thus, $\left(x,y\right)=\left(3,1\right)$

(ii) 2x + y = 5 .....(I)
3x $-$ y = 5 .....(II)
Adding (I) and (II)
$$\begin{gathered} 5x=10 \\ \Rightarrow x=\frac{10}{5}=2 \end{gathered}$$
Putting this value in (I) we have
$$\begin{gathered} 2\times 2+y=5 \\ \Rightarrow 4+y=5 \\ \Rightarrow y=1 \\ \left(x,y\right)=\left(2,1\right) \end{gathered}$$

(iii) 3x$-$ 5y =16 .....(I)
x $-$ 3y = 8 .....(II)
Multiply (II) with 3
$3x-9y=24$ .....(III)
(I) $-$ (III)
$$\begin{gathered} 4y=-8 \\ y=-2 \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}y\mathrm{in}\left(\mathrm{II}\right) \\ x=2 \end{gathered}$$

(iv) 2y $-$ x = 0 $\Rightarrow x=2y$ .....(I)
10x + 15y = 105 .....(II)
Taking out 5 common from (II)
$2x+3y=21$ .....(III)
From (I) and (III)
$$\begin{gathered} 2\left(2y\right)+3y=21 \\ \Rightarrow 4y+3y=21 \\ \Rightarrow 7y=21 \\ \Rightarrow y=3 \\ x=2\times 3=6 \end{gathered}$$

(v) 2x + 3y + 4 = 0 .....(I)
x$-$ 5y = 11 .....(II)
From (I) we have
2x + 3y = $-$4 .....(III)
Multiply (II) with 2
$2x-10y=22$ .....(IV)
From (III) and (IV) we have
2x + 3y = $-$4
$2x-10y=22$
(III) $-$ (IV)
$$\begin{gathered} 13y=-26 \\ \Rightarrow y=-2 \end{gathered}$$
x = 11 + 5y
x = 1

(vi) 2x $-$7y = 7 .....(I)
3x + y = 22 .....(II)
Multiply (II) with 7
21x + 7y = 154 .....(III)
(I) + (III)
23x = 161
x = 7, y = 1