RS Aggarwal Solutions Class 10 Chapter 3 - Linear Equations in two variables (Ex 3B) Exercise 3.2

Before applying any solving method to questions in RS Aggarwal Chapter 3, it is best practice to write the equations in their standard form. The general form for a pair of linear equations in two variables, x and y, is:

• a₁x + b₁y + c₁ = 0

• a₂x + b₂y + c₂ = 0

Here, a₁, b₁, c₁, a₂, b₂, and c₂ are real numbers, and both a and b are not zero in their respective equations. Arranging the equations this way helps in consistently applying methods like substitution and elimination and in checking the conditions for unique, no, or infinite solutions.

Verifying your solution is a crucial step to ensure accuracy. To check if your values for x and y are correct, you must substitute them back into both original equations. Your solution is correct only if it satisfies both equations simultaneously, meaning the Left-Hand Side (LHS) equals the Right-Hand Side (RHS) for each one. If the values only work for one of the two equations, there has been a calculation error.

A very common mistake when using the substitution method is an error in sign handling. When you substitute an expression like (3 - 2x) for y in an equation like 4x - 5y = 10, you must multiply the entire expression by -5. Students often forget to distribute the negative sign correctly, writing 4x - 5(3 - 2x) and then incorrectly calculating it as 4x - 15 - 10x instead of the correct 4x - 15 + 10x. Always use parentheses during substitution to avoid this error.

The outcome of the substitution process reveals the nature of the solution:

• Unique Solution: The method will yield specific numerical values for both x and y. This happens when the lines represented by the equations intersect at a single point.

• No Solution: After substitution, the variables will cancel out, leaving a false statement (e.g., 5 = 9). This indicates that the lines are parallel and never intersect.

• Infinitely Many Solutions: The variables will cancel out, but you will be left with a true statement (e.g., 7 = 7). This means the equations represent the same line (coincident lines), and any point on the line is a solution.

The most critical first step in solving a word problem involving linear equations is to identify the unknown quantities and assign variables to them. Carefully read the problem to determine what you need to find. For example, if the problem involves prices of articles or ages of people, you would start by stating: 'Let the cost of one pen be ₹x and the cost of one notebook be ₹y.' Once the variables are clearly defined, you can translate the information from the problem into two distinct linear equations to be solved.

The fundamental difference lies in their approach to removing a variable:

• The substitution method works by replacing one variable. You solve one equation for one variable (e.g., y = mx + c) and substitute this entire expression into the other equation.

• The elimination method works by cancelling one variable. You manipulate one or both equations (by multiplying them by constants) so that the coefficients of one variable are equal and opposite. Then, you add the two equations together, causing that variable to be eliminated.

While both methods yield the same result, one may be more efficient than the other depending on the structure of the given equations.