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# Substitution Method

The solution of the simultaneous linear equations can be divided into two broad categories, Graphical Method, and Algebraic method. The substitution method is one of the categories of the algebraic method. In this article, you will learn what the substitution method is and how to solve the linear equation using the substitution method with examples.

## Substitution Method Definition

The substitution method is the algebraic method to solve simultaneous linear equations. As the word says, in this method, the value of one variable from one equation is substituted in the other equation. In this way, a pair of the linear equation gets transformed into one linear equation with only one variable, which can then easily be solved. Before moving to solve the linear equations using the substitution method, get an idea on what the algebraic method and graphical method is.

### Algebraic Method

An Algebraic method is a collection of several methods, which are used to solve a pair of the linear equations that includes two variables. Generally, the algebraic method can be sub-divided into three categories:

• Substitution method
• Elimination method
• Cross-multiplication method

### Graphical Method

The graphical method is also known as the geometric method and is used to solve the system of linear equations. In this method, the equations are designed based on the objective function and constraints. To solve the system of linear equations, this method has undergone different steps to obtain the solutions.

In this article, we will focus mainly on solving the linear equations using the first algebraic method called “Substitution Method” in detail.

## Substitution Method Steps

For instance, the system of two equations with two unknown values, the solution can be obtained by using the below steps. Here, the list of steps is provided to solve the linear equation. They are

• Simplify the given equation by expanding the parenthesis
• Solve one of the equations for either x or y
• Substitute the step 2 solution in the other equation
• Now solve the new equation obtained using elementary arithmetic operations
• Finally, solve the equation to find the value of the second variable

## Elimination by Substitution Method

In this method, the elimination of the variable can be performed by substituting the value of another variable in an equation. Hence, this method is called the elimination by substitution method.

Let us assume the system of linear equations

2x+3y = 13 and x-2y = -4

Given:

2x+3y = 13  …  (1)

x-2y = -4 …(2)

The equation (2) can be written as

x = 2y-4 … (3)

Now, in equation (1) eliminate the variable x by substituting the equation (3).

Hence, equation (1) becomes

2(2y-4) +3y = 13

Now, apply the distributive property for the above equation,

4y-8+3y = 13

Now, solve the above equation for the variable y

7y – 8 = 13

7y = 13+8

7y = 21

y= 21/7

y= 3

Hence, the value of y is 3.

Now, substituting y=3 in the equation (2), we get

x- 2(3) = -4

x – 6=-4

x = -4+6

x = 2

Therefore, the value of x is 2.

Hence, the solution for the system of linear equations is:

x = 2 and y=3

To check whether the obtained solution is correct or not, substitute the values of x and y in any of the given equations.

Verification:

Use Equation (2) to verify the solution

x-2y = -4

Now, substitute x= 2 and y=3

2-2(3) = -4

2-6= -4

-4=-4

Here, L.H.S = R.H.S

Hnece, the obtained solution is correct.

### Difference Between Substitution Method and Elimination method

As we know that the substitution method is the process of solving the equation to find the variable value, and the value is substituted in the other equation. In contrast, the elimination method is the process of eliminating the variables in the equation so that the system of the equation can be left as the function of a single variable.

So, the major difference between the substitution and elimination method is that the substitution method is the process of replacing the variable with a value, whereas the elimination method is the process of removing the variable from the system of linear equations.

### Substitution Method Examples

Example 1:

Solve 2x + 3y = 9 and x – y = 3

Solution:

Given:2x + 3y = 9 and x – y = 3

For solving simultaneous equations,

Let, 2x + 3y = 9……..(1)

and x – y = 3 ……..(2)

From Equation (2) we get,

y = x – 3……………(3)

Now, in the substitution method, we find the value of one variable in terms of others and then substitute back.

Now, we know that y = x – 3

Substituting the value of y in equation (1), we get

2x + 3y = 9

⇒ 2x + 3(x – 3) = 9

⇒ 2x + 3x – 9 = 9

⇒ 5x = 18

⇒ x =

$$\begin{array}{l} \frac {18}{5}\end{array}$$

Now, the value of y can be found out using equation (3).

So, y = x – 3

⇒ y =

$$\begin{array}{l} \frac {18}{5}\end{array}$$
– 3

⇒ y =

$$\begin{array}{l} \frac {3}{5}\end{array}$$

Hence the solution of simultaneous equation will be: x =

$$\begin{array}{l} \frac {18}{5}\end{array}$$
and y =
$$\begin{array}{l} \frac {3}{5}\end{array}$$

In this way, we can find out the value of the unknown variables x and y using the substitution method.

Example 2:

Solve the pair of linear equations: 4x + 6y = 10 and 2x – 3y = 8 using Substitution method.

Solution:

4x + 6y = 10 ………….(i)

2x – 3y = 8  ……………(ii)

Finding the value of y in terms of x from equation (1), we get-

4x + 6y = 10

⇒ 6y = 10 – 4x

⇒ y =

$$\begin{array}{l} \frac {10-4x}{6} \end{array}$$
……………….(3)

Using this method, substituting the value of y in equation (2), we get-

$$\begin{array}{l} \large 2x – 3 \left( \frac{10~-~4x}{6} \right) \end{array}$$
= 10

⇒ 2x – 5 + 2x = 10

⇒ 4x = 15

⇒ x =

$$\begin{array}{l} \frac {15}{4} \end{array}$$

Finding the value of y, substitute the value of x in equation (3), we get-

y =

$$\begin{array}{l}\large \frac {10~-~4*\left( \frac {15}{4} \right)}{6}\end{array}$$

⇒ y =

$$\begin{array}{l} \frac {10~-~15}{6}\end{array}$$

⇒ y =

$$\begin{array}{l} \frac {-5}{6} \end{array}$$

Hence the value of y is

$$\begin{array}{l} – \frac 56 \end{array}$$
and x is
$$\begin{array}{l} \frac {15}{4} \end{array}$$

Substitution method is generally used for solving simultaneous equations, which is relatively easy. There are direct methods like cross-multiplication methods which can directly give you the value of the unknown variables. Still, for simple equations, not involving hectic calculations, this method can be preferred over other algebraic methods- Elimination method and cross-multiplication method.

If the pair of linear equations has no solution, then after substitution you won’t get the same value of LHS and RHS. In the case of infinite solutions, both sides of the equation will be equal to the same constant.

You will get a unique solution only when you get a proper value of the unknown variable after substitution.

## Frequently Asked Question on the Substitution Method

### What is meant by the substitution method?

In mathematics, the substitution method is generally used to solve the system of equations. In this method, first, solve the equation for one variable, and substitute the value of the variable in the other equation.

### Mention the different methods to solve the system of equations linear equations in two variables.

The three methods to solve the system of linear equations in two variables are:
Substitution method
Elimination method
Cross-multiplication method

### Write down the steps involved in the substitution method.

The steps involved in the substitution method are:
Solve one of the equations either for the variable x or y
Now, substitute the solution from step 1 to the other equation
Finally, solve the equation to find the value of the other variable

### What is the benefit of using the substitution method?

The benefit of using the substitution method is that this method gives the exact values for the variables (x and y), which correspond to the point of intersection.

### Can the substitution method be used to solve the system of equations in three variables?

Generally, while solving the system of equations with three variables, either we can use the substitution method or the elimination method to make the system into the system of two equations with two variables.

To learn about other methods to solve linear equations in two variables, download BYJU’S – The Learning App. Happy Learning!

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