NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 4(iv)
NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables
Question:
Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
\(2x - 2y - 2 = 0\), \(4x - 4y - 5 = 0\)
Given:
Pair of linear equations:
1) \(2x - 2y - 2 = 0\)
2) \(4x - 4y - 5 = 0\)
To Find:
- Are the given pair of equations consistent or inconsistent?
- If consistent, obtain the solution graphically.
Formula:
- General form of a linear equation: \(ax + by + c = 0\)
- To check consistency for equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\):
If \(\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}\), the system is consistent and has a unique solution.
If \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}\), the system is inconsistent (no solution).
If \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\), the system is consistent with infinitely many solutions.
Solution:
Step 1: Write the equations in standard form.
$$ \begin{align*} 2x - 2y - 2 &= 0 \implies 2x - 2y = 2 \\ 4x - 4y - 5 &= 0 \implies 4x - 4y = 5 \end{align*} $$Step 2: Identify coefficients for comparison.
For \(2x - 2y = 2\): \(a_1 = 2\), \(b_1 = -2\), \(c_1 = 2\)
For \(4x - 4y = 5\): \(a_2 = 4\), \(b_2 = -4\), \(c_2 = 5\)
Step 3: Find the ratios and check for consistency.
$$ \frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2} \\ \frac{b_1}{b_2} = \frac{-2}{-4} = \frac{1}{2} \\ \frac{c_1}{c_2} = \frac{2}{5} $$Since \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}\), the system is inconsistent.
Step 4: Interpret graphically.
Rewrite each equation in slope-intercept form:
$$ \begin{align*} 2x - 2y = 2 &\implies x - y = 1 \implies y = x - 1 \\ 4x - 4y = 5 &\implies x - y = \frac{5}{4} \implies y = x - \frac{5}{4} \end{align*} $$Both lines have the same slope (1) but different intercepts. Therefore, they are parallel and do not intersect.
Result:
The given pair of linear equations is inconsistent. The lines are parallel and do not intersect.
Therefore, there is no solution to the given pair of equations.
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