Problem & Solution Of Simultaneous Linear Equations –
5 6 3 2
Example.1) Solve ------- + ------- = 4, -------- - ------- = 6
x + y x – y x + y x – y
Ans.) The given equations ara –
2 6
--------- + --------- = 4 …………………. (i)
x + y x – y
3 2
--------- - --------- = 6 …………………… (ii)
x + y x – y
1 1
putting, -------- = u, --------- = v, and we get –
x + y x – y
so, 2u + 6v = 4 …………………….(iii)
3u – 2v = 6 ……………..…….. (iv)
Now, we will multiply 3 in (iv), and we get –
9u – 6v = 18 ………………..(v)
Now, we will add (iii) & (v), and we will get –
2u + 6v = 4
9u – 6v = 18
-------------------
11u = 22
=> u = 2
1 2
=> ---------- = ---------
x + y 1
=> 2x + 2y = 1 …………….. (vi)
now, we will substitute u in (iii)
3u – 2v = 6
=> (3 X 2) – 2v = 6
=> 6 – 2v = 6
=> 2v = 0
=> v = 0
1
=> --------- = 0
x - y
=> x – y = 0 ……………(vii)
Now, we will multiply 2 in (vii), and we get –
2x – 2y = 0 ……………. (viii)
Now, we will add (vi) & (viii), and get –
2x + 2y = 1
2x – 2y = 0
----------------
4x = 1
=> x = 1/4
Substitute the value of x in (vii), and we get –
x – y = 0
=> 1/4 – y = 0
=> y = 1/4
Hence, x = 1/4, y = 1/4 is the solution of the given equations. (Ans.)
When the given equations in x & y are such that the coefficients of x & y in one equation are interchanged in the other. In such cases, once we add them and once we subtract them to obtain them in the form of x + y = a and x – y = b. These equations can now be easily solved.
Example.) Solve => 37x + 41y = 70, 41x + 37y = 86
Ans.) The given equations are –
37x + 41y = 70 ………………….. (i)
41x + 37y = 86 …………………….(ii)
Adding (i) & (ii) we get –
37x + 41y = 70
41x + 37y = 86
--------------------
78x + 78y = 156
=> 78 (x + y) = 156
=> x + y = 2 …………..(iii)
Subtracting (ii) from (i), and we get –
41x + 37y = 86
37x + 41y = 70
------------------
4x – 4y = 16
=> 4(x – y) = 16
=> x – y = 4 ………………... (iv)
Adding, (iii) & (iv) and we get –
x + y = 2
x – y = 4
---------------
2x = 6
=> x = 3
Now, we will substitute x in (iv), and we get –
x – y = 4
=> 3 – y = 4
=> y = - 1
Hence, x = 3 and y = -1 is the solutions of the given equations. (Ans.)