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Algebraic solution by substitution

The substitution method solves one equation for one variable and substitutes that expression into the other equation.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    The substitution method solves one equation for one variable and substitutes that expression into the other equation.

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

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

Definition

The substitution method solves one equation for one variable and substitutes that expression into the other equation.

Example

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

Rule to remember

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

Memory hook

First isolate, then replace.

Examples and method

Worked example

From x = y + 3 and 2x + y = 13, substitute x = y + 3 into the second equation. Then 2(y + 3) + y = 13, so 3y = 7 and y = 7/3. Now x = 16/3.

Method to apply

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

Diagram support

No special diagram is needed, but a box showing the substituted expression can help students track each step.

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

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

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.

Answer writing and exam use

1-mark answer

The substitution method solves one equation for one variable and substitutes that expression into the other equation.

2-mark answer

The substitution method solves one equation for one variable and substitutes that expression into the other equation. If x = expression or y = expression in one equation, substitute that expression into the other equation. If x = 2y + 1 from the first equation, put that value into the second equation to reduce the pair to one equation.

3-mark answer

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. If x = expression or y = expression in one equation, substitute that expression into the other equation. From x = y + 3 and 2x + y = 13, substitute x = y + 3 into the second equation. Then 2(y + 3) + y = 13, so 3y = 7 and y = 7/3. Now x = 16/3. The exam may ask you to solve a pair by substitution or identify the equation that should be substituted first. A common wrong answer is to substitute a value before fully isolating the variable. That creates algebra errors and wrong signs.
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