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Mathematics · 2023 · 3 marks
CBSE 2023 · Region 2 · Set 1 · Q30
(a) If the system of linear equations 2 x +3 y =7 and 2 a x +(a+b) y =28 have infinite number of solutions, then find the values of ' a ' and ' b '.
(a)
If the system of linear equations \[2 \mathrm{x}+3 \mathrm{y}=7 \text { and } 2 a \mathrm{x}+(a+b) \mathrm{y}=28 \] have infinite number of solutions, then find the values of ' $\displaystyle a$ ' and ' $\displaystyle b$ '.
OR
(b)
If $\displaystyle 217 \mathrm{x}+131 \mathrm{y}=913$ and \[131 \mathrm{x}+217 \mathrm{y}=827, \] then solve the equations for the values of x and y.
Marking-scheme solution
system has infinite number of solutions
\[\begin{aligned}
& \therefore \frac{2}{2 a}=\frac{3}{a+b}=\frac{7}{28} \\
& \Rightarrow \frac{1}{a}=\frac{1}{4} \Rightarrow a=4
\end{aligned}
\]
and \(\displaystyle \mathrm{a}+\mathrm{b}=12 \Rightarrow \mathrm{~b}=8\)
\(\displaystyle \left.\begin{array}{l}217 x+131 y=913
131 x+217 y=827\end{array}\right]\) Adding \(\displaystyle 348(x+y)=1740\)
\[x+y=5
\]
Subtracting, \(\displaystyle 86(\mathrm{x}-\mathrm{y})=86\)
\[\begin{aligned}
& x-y=1 \\
& \Rightarrow x=3, y=2
\end{aligned}
\]
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