Q1(ii):

Solve the following pair of linear equations by the elimination method and the substitution method : (ii) 3x + 4y = 10 and 2x – 2y = 2

Given: A pair of linear equations in two variables:

(1) $3x + 4y = 10$

(2) $2x - 2y = 2$

To Find: The values of $x$ and $y$ using both the Elimination Method and the Substitution Method.

Method 1: Elimination Method

Step 1: Align the coefficients. To eliminate $y$, we multiply equation (2) by $2$ so that the coefficients of $y$ in both equations have the same magnitude but opposite signs.

Equation (2) $\times 2$: $2(2x - 2y) = 2(2) \implies 4x - 4y = 4$ --- (3)

Step 2: Add the equations. Add equation (1) and equation (3) to eliminate $y$.

$(3x + 4y) + (4x - 4y) = 10 + 4$

$3x + 4x + 4y - 4y = 14$

$7x = 14$

Step 3: Solve for $x$.

$x = \frac{14}{7} = 2$

Step 4: Substitute $x$ into equation (2) to find $y$.

$2(2) - 2y = 2$

$4 - 2y = 2$

$-2y = 2 - 4$

$-2y = -2$

$y = 1$

Method 2: Substitution Method

Step 1: Express one variable in terms of the other. From equation (2):

$2x - 2y = 2$

Divide by 2: $x - y = 1 \implies x = y + 1$ --- (4)

Step 2: Substitute equation (4) into equation (1).

$3(y + 1) + 4y = 10$

Step 3: Solve for $y$.

$3y + 3 + 4y = 10$ [Using the Distributive Property]

$7y + 3 = 10$

$7y = 10 - 3$

$7y = 7$

$y = 1$

Step 4: Substitute $y = 1$ back into equation (4) to find $x$.

$x = 1 + 1$

$x = 2$

Verification:

Substitute $x=2, y=1$ into equation (1): $3(2) + 4(1) = 6 + 4 = 10$ (Correct).

Substitute $x=2, y=1$ into equation (2): $2(2) - 2(1) = 4 - 2 = 2$ (Correct).

Final Answer: $x = 2$ and $y = 1$