# On comparing the ratios $$a_1\over a_2$$, $$b_1\over b_2$$ and $$c_1\over c_2$$, find out whether the following pair of linear equations are consistent, or inconsistent.

Question : On comparing the ratios $$a_1\over a_2$$, $$b_1\over b_2$$ and $$c_1\over c_2$$, find out whether the following pair of linear equations are consistent, or inconsistent.

(i)  3x + 2y = 5; 2x – 3y = 7

(ii)  2x – 3y = 8; 4x – 6y = 9

(iii)  $$3\over 2$$x + $$5\over 3$$y = 7;  9x – 10y = 14

(iv)  5x – 3y = 11;  -10x + 6y = -22

(v)  $$4\over 3$$x + 2y = 8;  2x + 3y = 12

Solution :

(i)  Rewrite the given equations as:

3x + 2y – 5 = 0;  2x – 3y – 7 = 0

$$a_1$$ = 3, $$b_1$$ = 2, $$c_1$$ = 5

$$a_2$$ = 2, $$b_2$$ = -3, $$c_2$$ = -7

$$a_1\over a_2$$ = $$3\over 2$$,  $$b_1\over b_2$$ = $$2\over -3$$

Thus,   $$3\over 2$$  $$\ne$$  $$2\over -3$$,  i.e.  $$a_1\over a_2$$  $$\ne$$  $$b_1\over b_2$$

Hence, the pair of linear equations is consistent.

(ii)  Rewrite the given equations as:

2x – 3y – 8 = 0;  4x – 6y – 9 = 0

$$a_1$$ = 2, $$b_1$$ = -3, $$c_1$$ = -8

$$a_2$$ = 4, $$b_2$$ = -6, $$c_2$$ = -9

$$a_1\over a_2$$ = $$2\over 4$$ = $$1\over 2$$,  $$b_1\over b_2$$ = $$-3\over -6$$ = $$1\over 2$$, $$c_1\over c_2$$ = $$-8\over -9$$ = $$8\over 9$$

Thus,   $$1\over 2$$  = $$1\over 2$$ $$\ne$$  $$8\over 9$$,  i.e.  $$a_1\over a_2$$  = $$b_1\over b_2$$ $$\ne$$ $$c_1\over c_2$$

Hence, the pair of linear equations is inconsistent.

(iii)  Rewrite the given equations as:

$$3\over 2$$x + $$5\over 3$$y – 7 = 0;  9x – 10y – 14 = 0

$$a_1$$ = $$3\over 2$$, $$b_1$$ = $$5\over 3$$, $$c_1$$ = -7

$$a_2$$ = 9, $$b_2$$ = -10, $$c_2$$ = -14

$$a_1\over a_2$$ = $$1\over 6$$,  $$b_1\over b_2$$ = $$-1\over 6$$

Thus,   $$1\over 6$$  $$\ne$$  $$-1\over 6$$,  i.e.  $$a_1\over a_2$$  $$\ne$$  $$b_1\over b_2$$

Hence, the pair of linear equations is consistent.

(iv)  Rewrite the given equations as:

5x – 3y – 11 = 0;  -10x + 6y + 22 = 0

$$a_1$$ = 5, $$b_1$$ = -3, $$c_1$$ = -11

$$a_2$$ = -10, $$b_2$$ = 6, $$c_2$$ = 22

$$a_1\over a_2$$ = $$-1\over 2$$,  $$b_1\over b_2$$ = $$-1\over 2$$, $$c_1\over c_2$$ = $$-1\over 2$$

Thus,   $$-1\over 2$$ = $$-1\over 2$$ = $$-1\over 2$$,  i.e.  $$a_1\over a_2$$ = $$b_1\over b_2$$ = $$c_1\over c_2$$

Hence, the pair of linear equations is consistent (or dependent).

(v)  Rewrite the given equations as:

$$4\over 3$$x + 2y – 8 = 0;  2x + 3y – 12 = 0

$$a_1$$ = $$4\over 3$$, $$b_1$$ = 2, $$c_1$$ = -8

$$a_2$$ = 2, $$b_2$$ = 3, $$c_2$$ = -12

$$a_1\over a_2$$ = $$2\over 3$$,  $$b_1\over b_2$$ = $$2\over 3$$, $$c_1\over c_2$$ = $$2\over 3$$

Thus,   $$2\over 3$$ = $$2\over 3$$ = $$2\over 3$$,  i.e.  $$a_1\over a_2$$ = $$b_1\over b_2$$ = $$c_1\over c_2$$

Hence, the pair of linear equations is consistent (or dependent).