NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 3(iv)
NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 3(iv)
Question:
On comparing the ratios a1, b1, c1 and a2, b2, c2, find out whether the following pair of linear equations are consistent or inconsistent.
\( 5x - 3y = 11 \) ;
\( -10x + 6y = -22 \)
Given:
The pair of linear equations:
1) \( 5x - 3y = 11 \)
2) \( -10x + 6y = -22 \)
To Find:
Whether the given pair of 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 two linear equations in the form:
\( a_1x + b_1y + c_1 = 0 \)
\( a_2x + b_2y + c_2 = 0 \)
Compare the ratios:
\[
\frac{a_1}{a_2},\quad \frac{b_1}{b_2},\quad \frac{c_1}{c_2}
\]
Cases:
- Case 1: If \( \frac{a_1}{a_2} \ne \frac{b_1}{b_2} \), the system is consistent (has a unique solution).
- Case 2: If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \), the system is inconsistent (no solution).
- Case 3: If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \), the system is consistent (infinitely many solutions).
Solution:
Step 1: Write the equations in standard form \( a_1x + b_1y + c_1 = 0 \).
\[ \begin{align*} \text{Equation 1:} &\quad 5x - 3y = 11 \implies 5x - 3y - 11 = 0 \\ \text{Equation 2:} &\quad -10x + 6y = -22 \implies -10x + 6y + 22 = 0 \end{align*} \]Step 2: Identify the coefficients \( a_1, b_1, c_1 \) and \( a_2, b_2, c_2 \).
\[ \begin{align*} a_1 = 5,\quad b_1 = -3,\quad c_1 = -11 \\ a_2 = -10,\quad b_2 = 6,\quad c_2 = 22 \end{align*} \]Step 3: Calculate the ratios \( \frac{a_1}{a_2} \), \( \frac{b_1}{b_2} \), and \( \frac{c_1}{c_2} \).
\[ \frac{a_1}{a_2} = \frac{5}{-10} = -\frac{1}{2} \] \[ \frac{b_1}{b_2} = \frac{-3}{6} = -\frac{1}{2} \] \[ \frac{c_1}{c_2} = \frac{-11}{22} = -\frac{1}{2} \]Step 4: Compare the ratios.
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = -\frac{1}{2} \]All three ratios are equal.
Step 5: Conclusion based on the cases.
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \), the system is consistent and has infinitely many solutions.
Result:
The given pair of linear equations is consistent and has infinitely many solutions.
Comments
Post a Comment