NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 4(i)

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 4(i)

Question:

Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
\( x + y = 5 \),
\( 2x + 2y = 10 \)

Given:

The pair of linear equations:
1) \( x + y = 5 \)
2) \( 2x + 2y = 10 \)

To Find:

  • Whether the given pair of equations is consistent or inconsistent.
  • If consistent, obtain the solution graphically.

Formula:

  • For two linear equations \( a_1x + b_1y = c_1 \) and \( a_2x + b_2y = c_2 \):
    • If \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2} \), the equations are dependent and consistent (infinitely many solutions).
    • If \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2} \), the equations are inconsistent (no solution).
    • If \( \dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2} \), the equations are consistent and independent (unique solution).
  • Graphical solution: Draw both lines and observe their intersection.

Solution:

Step 1: Write the equations in standard form.

$$ \begin{align*} \text{Equation 1:} & \quad x + y = 5 \\ \text{Equation 2:} & \quad 2x + 2y = 10 \end{align*} $$

Step 2: Compare the coefficients.

$$ \begin{align*} a_1 = 1, \quad b_1 = 1, \quad c_1 = 5 \\ a_2 = 2, \quad b_2 = 2, \quad c_2 = 10 \\ \\ \dfrac{a_1}{a_2} = \dfrac{1}{2}, \quad \dfrac{b_1}{b_2} = \dfrac{1}{2}, \quad \dfrac{c_1}{c_2} = \dfrac{5}{10} = \dfrac{1}{2} \end{align*} $$

Since all three ratios are equal, the equations are dependent and consistent.

Step 3: Simplify the second equation.

$$ 2x + 2y = 10 \implies x + y = 5 $$

So, both equations represent the same line.

Step 4: Find points to draw the graph of \( x + y = 5 \).

  • If \( x = 0 \), \( y = 5 \) → Point \( (0, 5) \)
  • If \( y = 0 \), \( x = 5 \) → Point \( (5, 0) \)
  • If \( x = 2 \), \( y = 3 \) → Point \( (2, 3) \)

Plot these points and draw a straight line. Both equations give the same line.

Therefore, every point on the line \( x + y = 5 \) is a solution to the given pair of equations.

Result:

The given pair of equations is consistent and dependent.
There are infinitely many solutions.
Graphically, both equations represent the same line, so every point on the line \( x + y = 5 \) is a solution.

© Kaliyuga Ekalavya. All rights reserved.

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