(a) The difference between two numbers is 12. The greater number is 6 less than twice the smaller one.
(i) Representing the above situation, frame two linear equations in two variables.
(ii) Show that the equations have unique solution.
(iii) Solve the equations and hence find the numbers.
OR
(b) Solve the following equations graphically: x + y = 7 and 2x - 5y = 7
✅ Answer & Solution
**(a)** Let greater number $=x$, smaller number $=y$.
(i) $x-y=12$ ...(i); $x=2y-6$ i.e. $x-2y=-6$ ...(ii)
(ii) Here $\frac{a_1}{a_2}=\frac11=1$, $\frac{b_1}{b_2}=\frac{-1}{-2}=\frac12$. Since $\frac{a_1}{a_2}\ne\frac{b_1}{b_2}$, the equations have a **unique solution**.
(iii) Subtracting (ii) from (i): $(x-y)-(x-2y)=12-(-6) \Rightarrow y=18$. Then $x=12+18=30$.
The numbers are **30 and 18**.
**(b)** For $x+y=7$: points (0,7), (7,0), (3,4)
For $2x-5y=7$: points (1,-1), (6,1), (-4,-3)
Plotting both lines, they intersect at the point **(6, 1)**. So $x=6, y=1$ is the solution.
✅ Verified by Super Admin
