Substitution Method — Linear Equations

1. Introduction

The substitution method is a simple and effective way to solve a pair of linear equations in two variables. In this method, we solve one equation for one variable and substitute its value in the other equation. This reduces the pair to a single-variable equation, which we can solve easily.

This method is especially useful when one equation already has a variable alone (like \(x = 3 - y\) or \(y = 2x + 1\)).

2. Steps of the Substitution Method

To solve two equations using substitution, follow these steps:

2.1. Step 1: Solve One Equation for One Variable

Choose the equation that is easiest to rearrange.

Example:

From \(x + y = 7\), we can write: \(x = 7 - y\).

2.2. Step 2: Substitute into the Other Equation

Replace the chosen variable in the second equation using the expression found in Step 1.

Example:

Substitute \(x = 7 - y\) into the second equation.

2.3. Step 3: Solve the Single-Variable Equation

After substitution, the equation will have only one variable. Solve it normally.

2.4. Step 4: Substitute Back to Find the Other Variable

Once you get the value of one variable, substitute it back into any equation to find the other variable.

2.5. Step 5: Write the Final Solution as an Ordered Pair

The solution should be written in the form \((x, y)\).

3. Worked Examples

Let’s understand the substitution method with examples.

3.1. Example 1

Solve the system:

\(x + y = 10\)

\(x - y = 4\)

Step 1: From the first equation, express \(x\):

\(x = 10 - y\)

Step 2: Substitute this in the second equation:

\((10 - y) - y = 4\)

\(10 - 2y = 4\)

Step 3: Solve for \(y\):

\(-2y = -6\)

\(y = 3\)

Step 4: Substitute back to find \(x\):

\(x = 10 - 3 = 7\)

Solution: \((x, y) = (7, 3)\)

3.2. Example 2

Solve the system:

\(2x + y = 11\)

\(x = 3y - 2\)

Step 1: Second equation already gives \(x\).

Step 2: Substitute in first equation:

\(2(3y - 2) + y = 11\)

\(6y - 4 + y = 11\)

\(7y = 15\)

\(y = \dfrac{15}{7}\)

Step 3: Substitute to find \(x\):

\(x = 3y - 2 = 3 \times \dfrac{15}{7} - 2 = \dfrac{45}{7} - 2 = \dfrac{31}{7}\)

Solution: \((x, y) = (\dfrac{31}{7}, \dfrac{15}{7})\)

4. Special Cases

Sometimes special results occur while using substitution.

4.1. Case 1: No Solution (Parallel Lines)

You may get an equation like:

\(0 = 5\)

This is false, meaning the lines never meet and there is no solution.

4.2. Case 2: Infinite Solutions (Coincident Lines)