Q1(iii):

Solve the following pair of linear equations by the substitution method. (iii) 3x – y = 3; 9x – 3y = 9

Given: A pair of linear equations in two variables:

(1) $3x - y = 3$

(2) $9x - 3y = 9$

To Find: The solution $(x, y)$ for the given pair of linear equations using the substitution method.

Step 1: Express one variable in terms of the other using Equation (1).

From Equation (1):

$3x - y = 3$

Add $y$ to both sides:

$3x = 3 + y$

Subtract $3$ from both sides:

$y = 3x - 3$ --- (Equation 3)

Step 2: Substitute the expression for $y$ into Equation (2).

Equation (2) is $9x - 3y = 9$.

Substitute $y = 3x - 3$ into Equation (2):

$9x - 3(3x - 3) = 9$

Step 3: Solve the resulting equation.

Distribute the $-3$ across the terms inside the parentheses:

$9x - 9x + 9 = 9$

Combine the $x$ terms ($9x - 9x = 0$):

$0 + 9 = 9$

$9 = 9$

Step 4: Interpret the result.

[Since the variable $x$ has been eliminated and we have arrived at a true statement ($9 = 9$), this indicates that the two equations are dependent and represent the same line.]

Specifically, if we divide Equation (2) by $3$:

$\frac{9x}{3} - \frac{3y}{3} = \frac{9}{3}$

$3x - y = 3$

This is identical to Equation (1). Therefore, the pair of linear equations has infinitely many solutions.

Step 5: General form of the solution.

Since the equations are coincident, any value of $x$ will yield a corresponding value of $y$ that satisfies both equations. We can express the solution set as:

$y = 3x - 3$ for any real number $x$.

Final Answer: The pair of linear equations has infinitely many solutions, represented by the relation $y = 3x - 3$.