Find the solution of the pair of equations $\frac{x}{10}+\frac{y}{5}-1=0$ and $\frac{x}{8}+\frac{y}{6}=15$. Hence, find $λ$, if $y = λx + 5$.
Given:
The given pair of equations is $\frac{x}{10}+\frac{y}{5}-1=0$ and $\frac{x}{8}+\frac{y}{6}=15$ and $y = λx + 5$.
To do:
We have to solve the given system of equations and the value of $λ$.
Solution:
The given system of equations can be written as,
$\frac{x}{10}+\frac{y}{5}=1$
$\Rightarrow \frac{1(x)+2(y)}{10}=1$
$\Rightarrow x+2y=1(10)$ (On cross multiplication)
$\Rightarrow x+2y=10$---(i)
$\frac{x}{8}+\frac{y}{6}=15$
$\Rightarrow \frac{3(x)+4(y)}{24}=15$ (LCM of 8 and 6 is 24)
$\Rightarrow 3x+4y=15(24)$ (On cross multiplication)
$\Rightarrow 3x=360-4y$
$\Rightarrow x=\frac{360-4y}{3}$----(ii)
Substitute $x=\frac{360-4y}{3}$ in equation (i), we get,
$\frac{360-4y}{3}+2y=10$
Multiplying by $3$ on both sides, we get,
$3(\frac{360-4y}{3})+3(2y)=3(10)$
$360-4y+6y=30$
$2y=30-360$
$2y=-330$
$y=\frac{-330}{2}$
$y=-165$
Substituting the value of $y=-165$ in equation (ii), we get,
$x=\frac{360-4(-165)}{3}$
$x=\frac{360+660}{3}$
$x=\frac{1020}{3}$
$x=340$
$y = λx + 5$ (Given)
$-165=λ(340)+5$
$340λ=-165-5$
$λ=\frac{-170}{340}$
$λ=\frac{-1}{2}$
Therefore, the solution of the given system of equations is $x=340$, $y=-165$ and the value of $λ$ is $\frac{-1}{2}$.
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