Algebraic solution by substitution Class 10 Maths | Pair of Linear Equations in Two Variables

Understand it

This method is useful when one equation already gives a variable in simple form, such as x = ... or y = .... After replacing that variable in the second equation, we get a single equation in one variable. Then we solve and back-substitute to find the other variable.

Example

If x = 2y + 1 from the first equation, put that value into the second equation to reduce the pair to one equation.

Formula and rule

If x = expression or y = expression in one equation, substitute that expression into the other equation.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

From x = y + 3 and 2x + y = 13, substitute x = y + 3 into the second equation.

1Then 2(y + 3) + y = 13, so 3y = 7 and y = 7/3.
2Now x = 16/3.

Method you can reuse

1Isolate one variable in one equation.
2Substitute that expression into the other equation.
3Solve the single-variable equation.
4Substitute back to get the second variable.
5Check both values in both equations.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is substitution useful when one variable is already isolated in a pair of equations?

Think first, then show answer

It is useful because the isolated variable can be replaced directly in the other equation. That reduces two variables to one variable and makes the pair easier to solve.

Common wrong answer

A common wrong answer is to substitute a value before fully isolating the variable. That creates algebra errors and wrong signs.

Don't mix this up

Students often substitute into the same equation they used to isolate the variable, which gives no new information.

Exam tip

Choose substitution when one variable is already alone or can be made alone easily.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

The exam may ask you to solve a pair by substitution or identify the equation that should be substituted first.

1
Enough for 1 mark
The substitution method solves one equation for one variable and substitutes that expression into the other equation.
2
Add for 2 marks
Example: If x = 2y + 1 from the first equation, put that value into the second equation to reduce the pair to one equation.
3
Add for 3 marks
This method is useful when one equation already gives a variable in simple form, such as x = ... or y = ....

Use the concept graph around this idea instead of revising it in isolation.

Pair of linear equations in two variables

A pair of linear equations in two variables is two first-degree equations written together, such as ax + by + c = 0 and px + qy + r = 0, with the same two unknowns.

Checking obtained solution in both equations

Checking means substituting the obtained values of x and y into both equations to confirm that the pair is correct.

Algebraic solution by elimination

The elimination method solves a pair by making one variable cancel when the equations are added or subtracted.

Algebraic solution by cross-multiplication

Cross-multiplication is a method used to solve a pair of linear equations written in the form a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0.

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Algebraic solution by substitution

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See it worked out

Check yourself

Write it in the exam

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Algebraic solution by substitution

Understand it

See it worked out

Check yourself

Write it in the exam

Related concepts

Practice this concept with focused MCQs

Help improve this page