3. On comparing the ratios , and , find out whether the following pair of linear equations are consistent, or inconsistent. (i) ; (ii) ; (iii) ; (iv) ; (v) ;

Understand the Question A pair of linear equations in standard form and is: Consistent if it has at least one solution: Intersecting lines (unique solution): Coincident lines (infinitely many solutions): Inconsistent if it has no solution: Parallel lines (no solution): Part (i) ; Step 1 · Compare Ratios of Coefficients Writing equations in standard form Here, , , and , , . Calculating the ratios Comparing the ratios Answer > (i) Consistent Part (ii) ; Step 1 · Compare Ratios of Coefficients Writing equations in standard form Here, , , and , , . Calculating the ratios Comparing the ratios Answer > (ii) Inconsistent Part (iii) ; Step 1 · Compare Ratios of Coefficients Writing equations in standard form Here, , , and , , . Calculating the ratios Comparing the ratios Answer > (iii) Consistent Part (iv) ; Step 1 · Compare Ratios of Coefficients Writing equations in standard form Here, , , and , , . Calculating the ratios Comparing the ratios Answer > (iv) Consistent Part (v) ; Step 1 · Compare Ratios of Coefficients Writing equations in standard form Here, , , and , , . Calculating the ratios Comparing the ratios Answer > (v) Consistent Common Mistakes Sign Errors with Constants: Always move all terms to the LHS () so that the signs of and are evaluated consistently. Fraction Ratio Simplification: In part (iii), ensure that and note that due to the negative sign. Consistent vs Inconsistent: Both unique solutions (intersecting lines) and infinitely many solutions (coincident lines) are consistent. Only parallel lines with no solution are inconsistent.

Pair of Linear Equations in Two Variables | Exercise 3.1

Question 3

On comparing the ratios $\dfrac{a_1}{a_2}$ , $\dfrac{b_1}{b_2}$ and $\dfrac{c_1}{c_2}$ , 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) $\dfrac{3}{2}x + \dfrac{5}{3}y = 7$ ; $9x - 10y = 14$

(iv) $5x - 3y = 11$ ; $-10x + 6y = -22$

(v) $\dfrac{4}{3}x + 2y = 8$ ; $2x + 3y = 12$

Solution

Understand the Question

A pair of linear equations in standard form $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ is:

Consistent if it has at least one solution:
- Intersecting lines (unique solution): $\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$
- Coincident lines (infinitely many solutions): $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$
Inconsistent if it has no solution:
- Parallel lines (no solution): $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$

(i) $3x + 2y = 5$ ; $2x - 3y = 7$

Step 1 · Compare Ratios of Coefficients

Writing equations in standard form $ax + by + c = 0$ $3x + 2y - 5 = 0$ $2x - 3y - 7 = 0$

Here, $a_1 = 3$ , $b_1 = 2$ , $c_1 = -5$ and $a_2 = 2$ , $b_2 = -3$ , $c_2 = -7$ .

Calculating the ratios $\dfrac{a_1}{a_2} = \dfrac{3}{2}$ $\dfrac{b_1}{b_2} = \dfrac{2}{-3} = -\dfrac{2}{3}$

Comparing the ratios $\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$

Answer

(i) Consistent

(ii) $2x - 3y = 8$ ; $4x - 6y = 9$

Step 1 · Compare Ratios of Coefficients

Writing equations in standard form $ax + by + c = 0$ $2x - 3y - 8 = 0$ $4x - 6y - 9 = 0$

Here, $a_1 = 2$ , $b_1 = -3$ , $c_1 = -8$ and $a_2 = 4$ , $b_2 = -6$ , $c_2 = -9$ .

Calculating the ratios $\dfrac{a_1}{a_2} = \dfrac{2}{4} = \dfrac{1}{2}$ $\dfrac{b_1}{b_2} = \dfrac{-3}{-6} = \dfrac{1}{2}$ $\dfrac{c_1}{c_2} = \dfrac{-8}{-9} = \dfrac{8}{9}$

Comparing the ratios $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$

Answer

(ii) Inconsistent

(iii) $\dfrac{3}{2}x + \dfrac{5}{3}y = 7$ ; $9x - 10y = 14$

Step 1 · Compare Ratios of Coefficients

Writing equations in standard form $ax + by + c = 0$ $\dfrac{3}{2}x + \dfrac{5}{3}y - 7 = 0$ $9x - 10y - 14 = 0$

Here, $a_1 = \dfrac{3}{2}$ , $b_1 = \dfrac{5}{3}$ , $c_1 = -7$ and $a_2 = 9$ , $b_2 = -10$ , $c_2 = -14$ .

Calculating the ratios $\dfrac{a_1}{a_2} = \dfrac{\frac{3}{2}}{9} = \dfrac{3}{18} = \dfrac{1}{6}$ $\dfrac{b_1}{b_2} = \dfrac{\frac{5}{3}}{-10} = \dfrac{5}{-30} = -\dfrac{1}{6}$

Comparing the ratios $\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$

Answer

(iii) Consistent

(iv) $5x - 3y = 11$ ; $-10x + 6y = -22$

Step 1 · Compare Ratios of Coefficients

Writing equations in standard form $ax + by + c = 0$ $5x - 3y - 11 = 0$ $-10x + 6y + 22 = 0$

Here, $a_1 = 5$ , $b_1 = -3$ , $c_1 = -11$ and $a_2 = -10$ , $b_2 = 6$ , $c_2 = 22$ .

Calculating the ratios $\dfrac{a_1}{a_2} = \dfrac{5}{-10} = -\dfrac{1}{2}$ $\dfrac{b_1}{b_2} = \dfrac{-3}{6} = -\dfrac{1}{2}$ $\dfrac{c_1}{c_2} = \dfrac{-11}{22} = -\dfrac{1}{2}$

Comparing the ratios $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$

Answer

(iv) Consistent

(v) $\dfrac{4}{3}x + 2y = 8$ ; $2x + 3y = 12$

Step 1 · Compare Ratios of Coefficients

Writing equations in standard form $ax + by + c = 0$ $\dfrac{4}{3}x + 2y - 8 = 0$ $2x + 3y - 12 = 0$

Here, $a_1 = \dfrac{4}{3}$ , $b_1 = 2$ , $c_1 = -8$ and $a_2 = 2$ , $b_2 = 3$ , $c_2 = -12$ .

Calculating the ratios $\dfrac{a_1}{a_2} = \dfrac{\frac{4}{3}}{2} = \dfrac{4}{6} = \dfrac{2}{3}$ $\dfrac{b_1}{b_2} = \dfrac{2}{3}$ $\dfrac{c_1}{c_2} = \dfrac{-8}{-12} = \dfrac{2}{3}$

Comparing the ratios $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$

Answer

(v) Consistent

Common Mistakes

Sign Errors with Constants: Always move all terms to the LHS ( $ax + by + c = 0$ ) so that the signs of $c_1$ and $c_2$ are evaluated consistently.
Fraction Ratio Simplification: In part (iii), ensure that $\dfrac{3/2}{9} = \dfrac{3}{18} = \dfrac{1}{6}$ and note that $\dfrac{1}{6} \neq -\dfrac{1}{6}$ due to the negative sign.
Consistent vs Inconsistent: Both unique solutions (intersecting lines) and infinitely many solutions (coincident lines) are consistent. Only parallel lines with no solution are inconsistent.

Question 3

More questions in Exercise 3.1