Q1(iv):

Solve the following pair of linear equations by the substitution method. (iv) 0.2x + 0.3y = 1.3; 0.4x + 0.5y = 2.3

Given: A pair of linear equations in two variables:

(1) $0.2x + 0.3y = 1.3$

(2) $0.4x + 0.5y = 2.3$

To find: The values of $x$ and $y$ using the substitution method.

Step 1: Simplifying the equations

To make the calculations easier, we multiply both equations by $10$ to eliminate the decimals.

Equation (1) becomes: $10(0.2x + 0.3y) = 10(1.3) \implies 2x + 3y = 13$ --- (Equation 3)

Equation (2) becomes: $10(0.4x + 0.5y) = 10(2.3) \implies 4x + 5y = 23$ --- (Equation 4)

Step 2: Expressing one variable in terms of the other

From Equation (3), we isolate $x$:

$2x = 13 - 3y$

$x = \frac{13 - 3y}{2}$ --- (Equation 5)

Step 3: Substituting the expression into the second equation

Substitute the value of $x$ from Equation (5) into Equation (4):

$4\left(\frac{13 - 3y}{2}\right) + 5y = 23$

[Since $4/2 = 2$, we simplify the expression]

$2(13 - 3y) + 5y = 23$

Step 4: Solving for $y$

Distribute the $2$ across the terms in the parentheses:

$26 - 6y + 5y = 23$

[Combine like terms: $-6y + 5y = -y$]

$26 - y = 23$

Subtract $26$ from both sides:

$-y = 23 - 26$

$-y = -3$

$y = 3$

Step 5: Solving for $x$

Substitute the value $y = 3$ back into Equation (5):

$x = \frac{13 - 3(3)}{2}$

$x = \frac{13 - 9}{2}$

$x = \frac{4}{2}$

$x = 2$

Step 6: Verification (Optional but recommended)

Substitute $x=2$ and $y=3$ into the original Equation (1):

$0.2(2) + 0.3(3) = 0.4 + 0.9 = 1.3$ (Correct)

Substitute $x=2$ and $y=3$ into the original Equation (2):

$0.4(2) + 0.5(3) = 0.8 + 1.5 = 2.3$ (Correct)

Final Answer: $x = 2, y = 3$