NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 3(v)
NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables
Question:
On comparing the ratio, \( \left(\frac{a_1}{a_2}\right), \left(\frac{b_1}{b_2}\right), \left(\frac{c_1}{c_2}\right) \), find out whether the following pair of linear equations are consistent, or inconsistent.
\( \frac{4}{3}x + 2y = 8 \) ; \( 2x + 3y = 12 \)
Given:
The two linear equations are:
(1) \( \frac{4}{3}x + 2y = 8 \)
(2) \( 2x + 3y = 12 \)
To Find:
Whether the given pair of linear equations are consistent or inconsistent by comparing the ratios:
\( \left(\frac{a_1}{a_2}\right), \left(\frac{b_1}{b_2}\right), \left(\frac{c_1}{c_2}\right) \)
Formula:
For two linear equations in the form:
\( a_1x + b_1y + c_1 = 0 \)
\( a_2x + b_2y + c_2 = 0 \)
The consistency is checked as follows:
- Case 1: If \( \frac{a_1}{a_2} \ne \frac{b_1}{b_2} \), equations are consistent and have a unique solution.
- Case 2: If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \), equations are inconsistent and have no solution.
- Case 3: If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \), equations are consistent and have infinitely many solutions.
Solution:
Step 1: Write the equations in standard form
$$ \begin{align*} \text{Equation 1:} & \quad \frac{4}{3}x + 2y = 8 \implies \frac{4}{3}x + 2y - 8 = 0 \\ \text{Equation 2:} & \quad 2x + 3y = 12 \implies 2x + 3y - 12 = 0 \end{align*} $$Step 2: Identify the coefficients
$$ \begin{align*} a_1 = \frac{4}{3}, \quad b_1 = 2, \quad c_1 = -8 \\ a_2 = 2, \quad b_2 = 3, \quad c_2 = -12 \end{align*} $$Step 3: Calculate the ratios
$$ \frac{a_1}{a_2} = \frac{\frac{4}{3}}{2} = \frac{4}{3} \times \frac{1}{2} = \frac{4}{6} = \frac{2}{3} $$ $$ \frac{b_1}{b_2} = \frac{2}{3} $$ $$ \frac{c_1}{c_2} = \frac{-8}{-12} = \frac{8}{12} = \frac{2}{3} $$Step 4: Compare the ratios
$$ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = \frac{2}{3} $$Step 5: Conclusion
Since all three ratios are equal, by the formula, the equations are consistent and have infinitely many solutions.
Result:
The given pair of linear equations are consistent and have infinitely many solutions.
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