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# Q3. 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)$$\frac{4}{3}x + 2y = 8, 2x + 3y = 12$$

Ans.(i) $$3x + 2y = 5, 2x - 3y = 7$$
On comparing these equations with the general form: $$ax^2 + bx + c$$
$$a_1 = 3, b_1 = 2, c_1 = -5$$
$$a_2 = 2, b_2 = -3, c_2 = -7$$
Here $${{a_1}\over {a_2}} \ne {{b_1}\over {b_2}}$$ as,
$${{3}\over {2}} \ne {{2}\over {3}}$$
So, these lines have a unique solution which means they intersect at one point.
Hence, they are consistent.

(ii) $$2x - 3y = 8, 4x - 6y = 9$$
On comparing these equations with the general form: $$ax^2 + bx + c$$
$$a_1 = 2, b_1 = -3, c_1 = -8$$
$$a_2 = 4, b_2 = 6, c_2 = -9$$
Here $${{a_1}\over {a_2}} = {{b_1}\over {b_2}} \ne {{c_1}\over {c_2}}$$ as,
$${{1}\over {2}} = {{1}\over {2}} \ne {{-8}\over {-9}}$$
So, these lines are parallel to each other.
Hence, they are inconsistent.

(iii) $${{3}\over {2}}x + {{5}\over {3}}y = 7,9x - 10y = 14$$
On comparing these equations with the general form: $$ax^2 + bx + c$$
$$a_1 = \frac{3}{2}, b_1 = \frac{5}{3}, c_1 = 7$$
$$a_2 = 9, b_2 = -10, c_2 = 14$$
Here $${{a_1}\over {a_2}} \ne {{b_1}\over {b_2}}$$ as,
$${{{{3}\over {2}}}\over {9}} \ne {{{{5}\over {3}}}\over {-10}}$$
=>$${{1}\over {6}} \ne {{-1}\over {6}}$$
So, these lines have a unique solution which means they intersect at one point.
Hence, they are consistent.

(iv) $$5x - 3y = 11, -10x + 6y = -22$$
On comparing these equations with the general form: $$ax^2 + bx + c$$
$$a_1 = 5, b_1 = -3, c_1 = 11$$
$$a_2 = -10, b_2 = 6, c_2 = -22$$
Here $${{a_1}\over {a_2}} = {{b_1}\over {b_2}} = {{c_1}\over {c_2}}$$ as,
$${{5}\over {-10}} = {{-3}\over {6}} = {{11}\over {-22}}$$
=>$${{-1}\over {2}} = {{-1}\over {2}} = {{-1}\over {2}}$$
So, these lines have infinite many solutions.
Hence, they are consistent.

(v)$$\frac{4}{3}x + 2y = 8, 2x + 3y = 12$$
On comparing these equations with the general form: $$ax^2 + bx + c$$
$$a_1 = \frac{4}{3}, b_1 = 2, c_1 = 8$$
$$a_2 = 2, b_2 = 3, c_2 = 12$$
Here $${{a_1}\over {a_2}} = {{b_1}\over {b_2}} = {{c_1}\over {c_2}}$$ as,
$${{{{4}\over {3}}}\over {2}} = {{2}\over {3}} = {{8}\over {12}}$$
=>$${{2}\over {3}} = {{2}\over {3}} = {{2}\over {3}}$$
So, these lines have infinite many solutions.
Hence, they are consistent.