# CLASS-9PROBLEM & SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS

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 = 1/4y = 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.)