(x+7)/3-y=(y-1)/2+x=(x+y)1/6 Solve for x and y. Substitution Method [S. Chand ICSE Class 9 Chapter 4 Exercise 4.2 Simultaneous Linear Equations of Two Variables Solution of the Book Fundamentals of Mathematics]
Substitution Method
$\frac{x + 7}{3} – y = \frac{y – 1}{2} + x = \frac{1}{6}(x + y)$
∴ $\frac{x + 7}{3} – y = \frac{1}{6}(x + y)$ —(i)
and $\frac{y – 1}{2} + x = \frac{1}{6}(x + y)$ —(ii)
From Equation (i) we get,
$\frac{x + 7}{3} – y = \frac{1}{6}(x + y)$
$\Rightarrow \frac{x + 7 – 3y}{3}$ = $\frac{x + y}{6}$
$\Rightarrow \frac{x + 7 – 3y}{1}$ = $\frac{x + y}{2}$
$\Rightarrow 2(x + 7 – 3y)$ = x + y
$\Rightarrow 2x + 14 – 6y$ = x + y
$\Rightarrow 2x – x $= 6y + y – 14
$\Rightarrow x$ = 7y – 14 —(iii)
From equation (ii) Substituting the value of x we get,
$\frac{y – 1}{2} + x = \frac{x + y}{6}$
$\Rightarrow \frac{y – 1}{2} + 7y – 14 = \frac{7y – 14 + y}{6}$
$\Rightarrow \frac{y – 1 + 14y – 28}{2} = \frac{8y – 14}{6}$
$\Rightarrow \frac{15y – 29}{1} = \frac{8y – 14}{3}$
$\Rightarrow 45y – 87 = 8y – 14$
$\Rightarrow 45y – 8y = 87 – 14$
$\Rightarrow 37y = 73$
$\Rightarrow y = \frac{73}{37}$
From Equation (iii) we get,
${\mathrm{x}} = 7\left(\frac{73}{37}\right) – 14$
$\Rightarrow {\mathrm{x}} = \frac{511}{37} – 14$
$\Rightarrow {\mathrm{x}} = \frac{511 – 518}{37}$
$\Rightarrow {\mathrm{x}} = – \frac{7}{37}$
∴ Required Solutions are x =- $\frac{7}{37}$ and y = $\frac{73}{37}$
Elemination Method
$\frac{x + 7}{3} – y = \frac{y – 1}{2} + x = \frac{1}{6}(x + y)$
∴ $\frac{x + 7}{3} – y = \frac{1}{6}(x + y)$
and $\frac{y – 1}{2} + x = \frac{1}{6}(x + y)$ —(ii)
From Equation (i) we get,
$\frac{x + 7}{3} – y = \frac{1}{6}(x + y)$
$\Rightarrow \frac{x + 7 – 3y}{3}$ = $\frac{x + y}{6}$
$\Rightarrow \frac{x + 7 – 3y}{1}$ = $\frac{x + y}{2}$
$\Rightarrow 2(x + 7 – 3y)$ = x + y
$\Rightarrow 2x + 14 – 6y$ = x + y
$\Rightarrow 2x – x $= 6y + y – 14
$\Rightarrow x-7y = – 14$ —(iii)
From Equation (ii) we get,
$\frac{{\mathrm{y}} – 1}{2} + {\mathrm{x}} = \frac{{\mathrm{x}} + {\mathrm{y}}}{6}$
$\Rightarrow \frac{y – 1 + 2x}{2} = \frac{x + y}{6}$
$\Rightarrow y + 2x – 1 = \frac{x + y}{3}$
$\Rightarrow 3y + 6x – 3 = x + y$
$\Rightarrow 3y – y + 6x – x = 3$
$\Rightarrow 5x +2y = 3$ —-(iv)
Multiplying Equation (iii) with 2 and Equation (iv) with 7 we get,
2x -14y= -28 —-(v)
35x+14y=21 —-(vi)
Adding Equation (v) and (vi) we get,
37x = -7
⟹ x = $\frac{ – 7}{37}$
Substituting the value of x in equation (iii) we get,
$\frac{ – 7}{37} – 7{\mathrm{y}} = – 14$
$\Rightarrow – 7{\mathrm{y}} = \frac{7}{37} – 14$
$\Rightarrow – 7y = \frac{7 – 518}{37}$
$\Rightarrow – 7y = – \frac{511}{37}$
$\Rightarrow y = \frac{73}{37}$
∴ Required Solutions are x =- $\frac{7}{37}$ and y = $\frac{73}{37}$