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# Substitution|Definition & Meaning

## Definition

In an **algebraic** equation, **substitution** refers to the **process** of replacing the **variables,** which are **represented** by the letters, with the **corresponding** numerical values. After that, we are able to **compute** the **overall** worth of the **expression.**

The **substitution** method is one of the **algebraic approaches** that can be **utilized** in the **resolution** of a **linear equation** system involving two variables. Using this approach, the **number** of any of the **variables** may be **determined** by **isolating** that **variable** on one aspect of the **equation** and **extracting** all of the other terms from the opposite of the **equation. **

After that, we enter that value as a **substitution** within the **second** equation. **Finding** the answers to a set of **linear equations** using the **substitution method** is a **straightforward** process that **requires** only a few **easy steps.**

The **following** figure **explains** the **substitution** method **graphically.** Suppose that we want to **add** the **three** objects as shown in the **figure.** By **substituting** the values of **three** objects we end up with a value of **12.**

## The Definition of the Substitution Method

The **algebraic** solution to the **problem** of solving **simultaneous** linear **equations** is called the **substitution** method. In this approach, the value of a variable taken through one equation is **swapped** into the **second** equation, just as the name of the method **suggests.**

Doing so **converts** a **pair of linear equations** into a **single linear equation **containing just **one variable** that can then be **solved** in a **straightforward** manner.

Try to achieve a basic **understanding** of the algebraic technique and the **graphical** method before moving on to **solving linear equations** with the **substitution method.** This will prepare you for the next step.

## Using Algebraic Methods

**Two-variable** linear **equations** can be solved **using** a variety of **algebraic** techniques. The **algebraic approach** can be broken down into these **three** broad classes:

**Method**of**substitution.****Method**of**elimination.****Method**of cross**multiplication.**

## Graphical Methods

To solve a system of **linear equations,** one can utilize a technique that is also called the **geometric method,** and that is called the **graphical method.** In this approach, the **equations** are **formulated** in such a way as to take into **account** both the **optimal solution** and the **constraints.** This procedure has been carried out in a series of phases in order to find the **solutions** in order to solve the **linear equations system.**

## Steps in the Substitution Method

For **example,** the solution to the **system** of two **equations having** two **unknown** values can be **determined** by following the **procedures** that are shown below. This **section contains** a list of the **steps** that must be taken in **order** to solve the **linear equation.** They **represent:**

**Expanding**the**brackets**will help simplify the equation that has been presented.Find a**solution**for either x or y in one of the**equations.**- In the other
**equation, substitute**the**solution**from step 2 in its place. - Now, using only the most
**fundamental arithmetic procedures,**calculate the newly**derived**equation. - Finally, the
**equation**needs to be solved in order to**determine**the value of a**second**variable.

## The Distinctive Features of the Elimination and Substitution Methods

As we know, the **substitution** approach **entails** solving one **equation** for the value of the **variable** in question and then **substituting** that value into the **second** equation. On the other hand, the **elimination method** involves **removing unnecessary** variables from an **equation** until the system is a **function** of a single one. The **substitution approach** can be used to find the values of x and y at the **intersection point** with precision**.**

**Consequently,** the key **distinction between** the **substitution approach** and the **elimination method** is that the former involves **replacing** the variable with such a value, and the latter involves **erasing** the variable from the **linear equation** system.

### Important Remarks Regarding the Method of Substitution

- To begin using the
**substitution approach,**pick an**equation**that has a**coefficient**of one for at least one of the**variables,**and then solve for that variable. This will get you**started**with the**procedure**(with coefficient 1).**Because**of this, the**process**is**simplified.** - Before
**beginning**the replacement process**,**merge any**phrases**that are the same (if any). - After we have
**determined**the value of one variable, we may use either one of the**following**equations or even any equation we have**encountered**thus far to get the value of the**second variable.** **Using**the**substitution**approach to**solve**a**problem,**if we find any true statements as 3 = 3, 0 = 0, etc.,**when**solving the**problem,**then this**indicates**that the**system**has an endless number of solutions.- When solving a problem using the
**substitution approach**, if we encounter any**false statements**such as 3 = 2, 0 = 1, etc., this**indicates**that the**problem**does not have a**solution.**

## Graphical Explanation of Substitution

The **following** figure shows the **arithmetic operations** using the **substitution** method. By **solving** the **below** figure, we **end** up with a value of **14.**

In the **figure below,** you can see how the **substitution method** is used to do the math. The **rectangle** value is 10, and the **triangle** value is **220.** When we solve the figure below, we get the **number 230.**

## Numerical Examples Using Substitution

### Example 1

Find out what **10x** – **5y** equals when **x = 20 and y = 4.**

### Solution

**Given** that:

**x = 20**

**y = 4**

The **given equation** is:

**10x – 5y**

By **putting** the **values** in the above **equation,** we get:

**= 10(20) – 5(4)**

**= 200 – 5(4)**

**= 200 – 20**

By **subtracting,** we **get:**

= **180**

### Example 2

Find out what **20x** – 5y **equals** when x = **220** and y = **40.**

### Solution

**Given that:**

**x = 220**

**y = 40**

The given **equation** is:

**20x – 5y**

By **putting** the values in the **above** equation, we get:

**= 10(220) – 5(40)**

**= 2200 – 5(40)**

**= 2200 – 200**

By **subtracting,** we get:

**= 2000**

*All images/mathematical drawings were created with GeoGebra.*