Question

Solve the following simultaneous equations.

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

(ii) x $-$2 y = $-$1 ; 2x $-$ y = 7

(iii) x + y = 11 ; 2x $-$ 3 y = 7

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

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

(vi) x$-$ 2y = $-$2 ; x + 2y = 10

Answer 1

(i) 2x + y = 5 .....(I)
3x $-$ y = 5 .....(II)
(I) + (II)
$$\begin{gathered} 5x=10 \\ \Rightarrow x=2 \end{gathered}$$
Putting the value of x in (I)
$$\begin{gathered} 2\times 2+y=5 \\ \Rightarrow 4+y=5 \\ \Rightarrow y=1 \end{gathered}$$

(ii) x $-$2y = $-$1 .....(I)
2x $-$ y = 7 .....(II)
Multiply (I) with 2
$2x-4y=-2$ .....(III)
Subtracting (III) from (II)
$$\begin{gathered} -3y=-9 \\ \Rightarrow y=3 \end{gathered}$$
Putting the value of y in (I) we get
$$\begin{gathered} x-2\times 3=-1 \\ \Rightarrow x-6=-1 \\ \Rightarrow x=5 \end{gathered}$$

(iii) x + y = 11 .....(I)
2x $-$ 3y = 7 .....(II)
Multiply (I) with 3
$3x+3y=33$ .....(III)
Adding (II) and (III)
$$\begin{gathered} 5x=40 \\ \Rightarrow x=8 \end{gathered}$$
Putting the value of x in (I)
$$\begin{gathered} 8+y=11 \\ \Rightarrow y=3 \end{gathered}$$

(iv) 2x + y = $-$2 .....(I)
3x $-$ y = 7 .....(II)
Adding (I) with (II)
$$\begin{gathered} 5x=5 \\ \Rightarrow x=1 \end{gathered}$$
Putting the value of x in (I)
$$\begin{gathered} 2\times 1+y=-2 \\ \Rightarrow 2+y=-2 \\ \Rightarrow y=-4 \end{gathered}$$

(v) 2x $-$ y = 5 .....(I)
3x + 2y = 11 .....(II)
Multiplying (I) with 2 we get
$4x-2y=10$ .....(III)
Adding (II) with (III)
$$\begin{gathered} 7x=21 \\ \Rightarrow x=3 \end{gathered}$$
Putting the value of x in (I) we get
$$\begin{gathered} 2\times 3-y=5 \\ \Rightarrow 6-y=5 \\ \Rightarrow y=1 \end{gathered}$$

(vi) x $-$ 2y = $-$2 .....(I)
x + 2y = 10 .....(II)
Adding (I) with (II)
$$\begin{gathered} 2x=8 \\ \Rightarrow x=4 \end{gathered}$$
Putting the value of x in (I) we get
$$\begin{gathered} 4-2y=-2 \\ \Rightarrow -2y=-6 \\ \Rightarrow y=3 \end{gathered}$$