NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.3 Question 1(i)
NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.3 Question 1(i)
Question:
Solve the following pair of linear equations by the elimination method and the substitution method:
\( x + y = 5 \) and \( 2x - 3y = 4 \)
Given:
The two linear equations are:
\( x + y = 5 \) (1)
\( 2x - 3y = 4 \) (2)
To Find:
The values of \( x \) and \( y \) using:
- Elimination method
- Substitution method
Formula:
Elimination Method: Eliminate one variable by making the coefficients of that variable equal (or opposites) and adding or subtracting the equations.
Substitution Method: Express one variable in terms of the other using one equation, then substitute this value into the second equation to solve.
Solution:
Step 1: Write the given equations.
$$ \begin{align*} x + y &= 5 \quad \text{(1)} \\ 2x - 3y &= 4 \quad \text{(2)} \end{align*} $$Step 2: Multiply equation (1) by 3 to make the coefficients of \( y \) equal and opposite.
$$ 3(x + y) = 3 \times 5 \implies 3x + 3y = 15 \quad \text{(3)} $$Step 3: Add equation (2) and equation (3) to eliminate \( y \).
$$ \begin{align*} 3x + 3y &= 15 \\ 2x - 3y &= 4 \\ \hline (3x + 3y) + (2x - 3y) &= 15 + 4 \\ (3x + 2x) + (3y - 3y) &= 19 \\ 5x + 0 &= 19 \\ 5x &= 19 \\ x &= \frac{19}{5} \end{align*} $$Step 4: Substitute \( x = \frac{19}{5} \) in equation (1) to find \( y \).
$$ \begin{align*} x + y &= 5 \\ \frac{19}{5} + y &= 5 \\ y &= 5 - \frac{19}{5} \\ y &= \frac{25}{5} - \frac{19}{5} \\ y &= \frac{6}{5} \end{align*} $$Step 5: From equation (1), express \( y \) in terms of \( x \).
$$ x + y = 5 \implies y = 5 - x $$Step 6: Substitute \( y = 5 - x \) into equation (2) and solve for \( x \).
$$ \begin{align*} 2x - 3y &= 4 \\ 2x - 3(5 - x) &= 4 \\ 2x - 15 + 3x &= 4 \\ (2x + 3x) - 15 &= 4 \\ 5x - 15 &= 4 \\ 5x &= 4 + 15 \\ 5x &= 19 \\ x &= \frac{19}{5} \end{align*} $$Step 7: Substitute \( x = \frac{19}{5} \) in \( y = 5 - x \) to find \( y \).
$$ \begin{align*} y &= 5 - x \\ y &= 5 - \frac{19}{5} \\ y &= \frac{25}{5} - \frac{19}{5} \\ y &= \frac{6}{5} \end{align*} $$Result:
The solution to the given pair of equations is:
\[
x = \frac{19}{5}, \quad y = \frac{6}{5}
\]
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