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.