Solve 7/x+8/y=2 and 2/x+3/y=17 By Elimination and Substitution Method
Substitution method
Given Equations are,
$\frac{7}{{\mathrm{x}}} + \frac{8}{{\mathrm{y}}} = 2$ —(i)
$\frac{2}{{\mathrm{x}}} + \frac{3}{{\mathrm{y}}} = 17$ —(ii)
Let, $\frac{1}{{\mathrm{x}}}$=u and $\frac{1}{{\mathrm{y}}}$ = v
Then Equations (i) and (ii) Become,
7u+8v=2 —(iii)
and 2u+3v=17 —(iv)
From Equation (iii) we get,
7u= 2-8v
$\Rightarrow {\mathrm{u}} = \frac{2 – 8{\mathrm{v}}}{7}$ —(v)
Substituting the value of u in equation (iv) we get,
$2{\mathrm{u}} + 3{\mathrm{v}} = 17$
$\Rightarrow 2\left(\frac{2 – 8{\mathrm{v}}}{7}\right) + 3{\mathrm{v}} = 17$
$\Rightarrow \frac{4 – 16{\mathrm{v}}}{7} + 3{\mathrm{v}} = 17$
$\Rightarrow \frac{4 – 16{\mathrm{v}} + 21{\mathrm{v}}}{7} = 17$
$\Rightarrow 4 + 5{\mathrm{v}} = 119$
$\Rightarrow 5{\mathrm{v}} = 119 – 4$
$\Rightarrow 5{\mathrm{v}} = 115$
$\Rightarrow {\mathrm{v}} = \frac{115}{5}$
$\Rightarrow {\mathrm{v}} = 23$
From equation (v) we get,
${\mathrm{u}} = \frac{2 – 8{\mathrm{v}}}{7}$
$\Rightarrow {\mathrm{u}} = \frac{2 – 8(23)}{7}$
$\Rightarrow {\mathrm{u}} = \frac{2 – 184}{7}$
$\Rightarrow {\mathrm{u}} = \frac{ – 182}{7}$
$\Rightarrow {\mathrm{u}} = – 26$
$\therefore {\mathrm{x}} = – \frac{1}{26}$ and ${\mathrm{y}} = \frac{1}{23}$
Elimination Method
Given Equations are,
$\frac{7}{{\mathrm{x}}} + \frac{8}{{\mathrm{y}}} = 2$ —(i)
$\frac{2}{{\mathrm{x}}} + \frac{3}{{\mathrm{y}}} = 17$ —(ii)
Let, $\frac{1}{{\mathrm{x}}}$=u and $\frac{1}{{\mathrm{y}}}$ = v
Then Equations (i) and (ii) Become,
7u+8v=2 —(iii)
and 2u+3v=17 —(iv)
Multiplying Equation (iii) by 3 and Equation (iv) by 8 we get,
21u+24v =6 —(v)
16u+24v=136 —(vi)
Subtracting (v) from (vi) we get,
16u+24v – 21u-24v = 136 – 6
⟹ -5u = 130
⟹ u= -26
Substituting the value of u in equation (iv) we get,
7u+8v=2
⟹ 7(-26)+8v=2
⟹ – 182+8v=2
⟹ 8v= 2+182
⟹ 8v =184
⟹ v = 23
$\therefore {\mathrm{x}} = – \frac{1}{26}$ and ${\mathrm{y}} = \frac{1}{23}$