On comparing the ratios 1 1 2 2 , a b a b and 1 2 c c , 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 5 7 2 3 x y ; 9x – 10y = 14 (iv) 5x – 3y = 11 ; – 10x + 6y = –22 (v) 4 2 8 3 x y ; 2x + 3y = 12
Step-by-Step Solution
Key Concept: For a pair of linear equations \(a_1x+b_1y=c_1\) and \(a_2x+b_2y=c_2\):<br>- If \(\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}\), the equations represent the same line – they are <b>consistent (coincident)</b>.<br>- If \(\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}<br>eq\dfrac{c_1}{c_2}\), the lines are parallel and distinct – they are <b>consistent (distinct)</b>.<br>- If \(\dfrac{a_1}{a_2}<br>eq\dfrac{b_1}{b_2}\), the lines intersect at a unique point – they are <b>inconsistent</b> (no common solution).
### (i) \(3x+2y=5\) and \(2x-3y=7\)
- \(\dfrac{a_1}{a_2}=\dfrac{3}{2}=1.5\)
- \(\dfrac{b_1}{b_2}=\dfrac{2}{-3}= -\dfrac{2}{3}\)
- Since \(\dfrac{a_1}{a_2}
eq\dfrac{b_1}{b_2}\), the pair is inconsistent.
### (ii) \(2x-3y=8\) and \(4x-6y=9\)
- \(\dfrac{a_1}{a_2}=\dfrac{2}{4}=\dfrac{1}{2}\)
- \(\dfrac{b_1}{b_2}=\dfrac{-3}{-6}=\dfrac{1}{2}\)
- \(\dfrac{c_1}{c_2}=\dfrac{8}{9}
eq\dfrac{1}{2}\)
- Here \(\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}
eq\dfrac{c_1}{c_2}\); therefore the equations are consistent (distinct) (parallel lines).
### (iii) \(\dfrac{3}{5}x+\dfrac{7}{2}y=3\) and \(9x-10y=14\)
- \(\dfrac{a_1}{a_2}=\dfrac{\frac{3}{5}}{9}=\dfrac{3}{45}=\dfrac{1}{15}\)
- \(\dfrac{b_1}{b_2}=\dfrac{\frac{7}{2}}{-10}= -\dfrac{7}{20}\)
- Since \(\dfrac{a_1}{a_2}
eq\dfrac{b_1}{b_2}\), the pair is inconsistent.
### (iv) \(5x-3y=11\) and \(-10x+6y=-22\)
- \(\dfrac{a_1}{a_2}=\dfrac{5}{-10}= -\dfrac{1}{2}\)
- \(\dfrac{b_1}{b_2}=\dfrac{-3}{6}= -\dfrac{1}{2}\)
- \(\dfrac{c_1}{c_2}=\dfrac{11}{-22}= -\dfrac{1}{2}\)
- All three ratios are equal; hence the equations are consistent (coincident) (the same line).
### (v) \(\dfrac{4}{2}x+\dfrac{8}{3}y=\text{(constant)}\) and \(2x+3y=12\)
- Simplify the first coefficients: \(\dfrac{4}{2}=2\).
- \(\dfrac{a_1}{a_2}=\dfrac{2}{2}=1\)
- \(\dfrac{b_1}{b_2}=\dfrac{\frac{8}{3}}{3}=\dfrac{8}{9}
eq1\)
- Since \(\dfrac{a_1}{a_2}
eq\dfrac{b_1}{b_2}\), the pair is inconsistent.
Summary of results
- (i) Inconsistent
- (ii) Consistent (distinct)
- (iii) Inconsistent
- (iv) Consistent (coincident)
- (v) Inconsistent
Correct Answer: (i) Inconsistent; (ii) Consistent (distinct); (iii) Inconsistent; (iv) Consistent (coincident); (v) Inconsistent