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    NCERT| RBSE, CBSE | CLASS TENTH, SUBJECT MATHS, UNIT SECOND, Algebraic Methods of Solving a Pair of Linear Equations (1) Substitution Method (ii) Elimination Method.

    explanation of the algebraic methods for solving a pair of linear equations:

    1. Substitution Method:

    Steps:

    1. Solve one equation for one variable: Choose an equation and rearrange it to isolate one variable in terms of the other.
    2. Substitute: Substitute the expression you found for that variable into the other equation. This will create an equation with only one variable.
    3. Solve: Solve the resulting equation for the remaining variable.
    4. Back-substitute: Plug the value you found for that variable back into the first equation to solve for the other variable.

    Example:

    Given the equations:

    2x + 3y = 7 x – y = 1

    Solve for x using substitution:

    • Rearrange the second equation to isolate x: x = y + 1
    • Substitute this expression for x in the first equation: 2(y + 1) + 3y = 7
    • Simplify and solve for y: 2y + 2 + 3y = 7 => 5y = 5 => y = 1
    • Substitute y = 1 back into x = y + 1 to find x: x = 1 + 1 => x = 2

    2. Elimination Method:

    Steps:

    1. Make coefficients equal: Multiply one or both equations by appropriate numbers so that one variable has the same coefficient (but opposite signs) in both equations.
    2. Add or subtract equations: Add or subtract the equations to eliminate one variable.
    3. Solve for remaining variable: Solve the resulting equation for the remaining variable.
    4. Back-substitute: Plug the value you found for that variable back into one of the original equations to solve for the other variable.

    Example:

    Given the same equations as before:

    2x + 3y = 7 x – y = 1

    Solve for x using elimination:

    • Multiply the second equation by 2: 2x – 2y = 2
    • Add the first and second equations: (2x + 3y) + (2x – 2y) = 7 + 2 => 4x + y = 9
    • Solve for x: 4x = 9 – y => x = (9 – y)/4
    • Substitute y = 1 (from the previous example) to find x: x = (9 – 1)/4 => x = 2

    Choosing the Method:

    • Substitution: Often simpler when one equation is already solved for one variable.
    • Elimination: Convenient when coefficients of one variable are already additive inverses or easily made so.

    Consider factors like personal preference and the ease of manipulating the given equations when selecting a method.