Homeclass 10MathsPair of Linear Equations in Two VariablesSubstitution Method

Substitution Method

The substitution method is an algebraic technique used to solve a pair of linear equations in two variables. It involves finding the value of one variable in terms of the other from one equation and substituting this expression into the second equation. This reduces the system to a single linear equation with one variable, which can be easily solved.

Key Concepts

First, select either of the two equations and isolate one variable, expressing it entirely in terms of the second variable and the constant terms.
Next, substitute this newly formed expression into the other original equation, creating a simple linear equation in one variable.
Finally, solve for this single variable and substitute its numerical value back into the rearranged first equation to find the exact value of the other variable.

Formula & Equation

Given a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, isolate x in equation 1 as x = -(b1y + c1)/a1, then substitute this into equation 2: a2(-(b1y + c1)/a1) + b2y + c2 = 0.

x and y are the unknown variables, while a1, b1, a2, b2 are the coefficients and c1, c2 are the constant terms.

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Common Misconceptions

Myth: Substituting the derived expression for a variable back into the exact same equation it was extracted from.
Fact: The expression must be substituted into the other equation in the pair. Substituting it into the same equation will cancel out all variables and result in a tautology like 0 = 0, which does not help find the solution.

Real World Applications

Determining the fixed charge of a taxi and the charge per kilometer when given the total fare for two different journey distances.
Calculating the individual ages of a father and son based on two statements comparing their ages at different points in time.

Frequently Asked Questions

How do I know if a pair of linear equations has no solution using the substitution method?
If substituting the variable results in a mathematically false statement without variables, such as 5 = 0, the equations represent parallel lines and have no solution.
Which variable should I choose to isolate first in the substitution method?
For simplicity and to avoid complex fractions, always try to isolate a variable that has a coefficient of 1 or -1 in either of the given equations.

Keep Learning

Algebraic Solution MethodElimination MethodPair of Linear Equations