NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 3(iii)

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 3(iii)

Question:

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

\[ \frac{3}{2}x + \frac{5}{3}y = 7; \] \[ 9x - 10y = 14. \]

Given:

The two linear equations are:

(1) \( \frac{3}{2}x + \frac{5}{3}y = 7 \)
(2) \( 9x - 10y = 14 \)

To Find:

Whether the given pair of linear equations is consistent or inconsistent by comparing the ratios: \[ \frac{a_1}{a_2}, \quad \frac{b_1}{b_2}, \quad \frac{c_1}{c_2} \]

Formula:

For a pair of linear equations:

\[ a_1x + b_1y + c_1 = 0 \] \[ a_2x + b_2y + c_2 = 0 \]
  • If \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the equations are consistent and have a unique solution.
  • If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \), the equations are inconsistent (no solution).
  • If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \), the equations are consistent and have infinitely many solutions.

Solution:

Step 1: Write the equations in standard form \( a_1x + b_1y + c_1 = 0 \).

\[ \frac{3}{2}x + \frac{5}{3}y = 7 \implies \frac{3}{2}x + \frac{5}{3}y - 7 = 0 \] \[ 9x - 10y = 14 \implies 9x - 10y - 14 = 0 \]

Step 2: Identify the coefficients.

\[ \begin{align*} a_1 &= \frac{3}{2}, \quad b_1 = \frac{5}{3}, \quad c_1 = -7 \\ a_2 &= 9, \quad b_2 = -10, \quad c_2 = -14 \end{align*} \]

Step 3: Calculate the ratios \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \), and \( \frac{c_1}{c_2} \).

\[ \frac{a_1}{a_2} = \frac{\frac{3}{2}}{9} = \frac{3}{2} \times \frac{1}{9} = \frac{3}{18} = \frac{1}{6} \] \[ \frac{b_1}{b_2} = \frac{\frac{5}{3}}{-10} = \frac{5}{3} \times \frac{1}{-10} = \frac{5}{-30} = -\frac{1}{6} \] \[ \frac{c_1}{c_2} = \frac{-7}{-14} = \frac{1}{2} \]

Step 4: Compare the ratios.

\[ \frac{a_1}{a_2} = \frac{1}{6}, \quad \frac{b_1}{b_2} = -\frac{1}{6}, \quad \frac{c_1}{c_2} = \frac{1}{2} \] Clearly, \[ \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \]

Step 5: Conclusion.

Since \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the given pair of linear equations is consistent and has a unique solution.

Result:

The given pair of linear equations is consistent (they have a unique solution).
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