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# Elimination|Definition & Meaning

**Definition**

In order to** solve** a **system of linear equations**, one of the techniques is to **eliminate** the **variables**. The **equation** in **one variable** can be **obtained** by **adding or subtracting** the equations. This process is known as **Elimination**. We can **eliminate** **one variable** by **adding** the equation if one **variable’s coefficients** are the** same** and their **signs are opposite**. A similar method can be used if the **coefficients** are the **same** for one of the variables, and the **sign of the coefficients** is also the **same**.

**Conceptual Overview **

Figure 1 – Conceptual overview of Elimination Process

There are **many ways** of **solving linear equations** algebraically, but the **elimination method** is the most **widely used**. In the elimination method, **one variable** is **eliminated** using **basic arithmetic operations**, and the **equation is simplified** to **compute** the **value** of the **second variable**.** To find** the** value** of the **variable eliminated**, we can **use any equation** with the **value of the variable** eliminated.

**Suppose** there is a **set of equations** involving **two variables x and y**.In order to find the **value of x and y** we have to **perform elimination** to **get** the** desired values**. Similarly, if we **want to define** our **equation** in **only one variable** that also can be **done by elimination**. So in short **elimination convert** an **equation** involving **many variables to** only a **few or one variable**.

We **perform** various **operations** on a given set of equations for **performing elimination** like** first multiplying** the equation with **some constant** and then **applying addition** and **subtraction** operations on the equation. We can also** take** the **elimination proces**s as an aid to **convert equations** to a **compact form**.

**Steps Involved in Elimination Process**

Here we will discuss **step by step guide** to the elimination process.

**Step 1**

**First of all** according to the given data **mark the set of equations as 1 or 2** or 3 depending upon the given situation.

Figure 2 – Marking Numbers to equations

**Step 2**

After numbering **multiply or divide** either of the **one equation** with such type of a **constant** that will **result in making** the equation’s **first term equivalent to the first term** of the **second equation**, which will eventually **help in getting** **one variable eliminated**.

Normally, we should **make** that** term equal whose variable** is **to be eliminated** so **if** we want to **eliminate the x term** we will **make the x term** of **both equations** **equal** and if we want to **eliminate the y term** of the equation we will **make the y term equal**.

Figure 3 – Multiplying equation with constant

**Step 3**

When we are **able to make** the **first term** of both equations **equal** by **multiplying or dividing**, the **next step** is to **perform arithmetic operations** either addition or subtraction. The **factor** that will **decide** **which operation** **to apply** to the equation **depends upon** the **fact** that **one variable must be eliminated** after performing that operation. It is totally a senior dependent.

Figure 4 – Applying Arithmetic Operation on Equation

**Step 4**

**After eliminating** one **variable**,** instead of repeating** the** whole process again,** we **can find the value** of **another unknown variable** by **just substituting** the **value of the first variable** we get in the equation. The **intuition of substitution** here is to just **get the answer** **quickly** while repeating the process for another variable would be time taking.

Figure 5 – Substituting Value of Variable Y in Equation

**General Tips for Elimination**

- If there are
**two equations**and we**want to eliminate one**of the**variables,**we will**add the equation’s first coefficients**if both**equations are different**.For instance, x+y=2, and 3x+y=1 - If the coefficient of
**both equations is equal,**we will s**ubtract the equations**. For instance, 2x+y=2, and 2x+3y=1 - In case one of the variables needs to be eliminated, the e
**limination method**is**preferable over substitution**. In order to solve the linear equation in one variable, the**previous step**is to**form the equation**in one variable.

**Elimination of Equation of Two Parallel Lines**

We know that **parallel lines** are **types of lines** that **never intersect each other.** So in order to show the **weakness of the elimination method,** we **will solve the equation of a parallel line** to show the results. In general, there is **no solution for two parallel lines**. Let’s take two parallel lines:

x+y=8 (Eq. 1)

2x+2y=8 (Eq. 2)

So in order to make the **first term of both equation equal,** we will **multiply equation one** **with 2** on each side, which will give:

2x+2y=16 (Eq. 3)

In order to eliminate we will **subtract equation 2 and equation 3:**

2x+2y-2x-2y=16-8

0=8

This is **wrong** as **zero can not be equal to 8,** which shows that we c**an not eliminate** any **variable of equations** that are **parallel**.

**Elimination of Equation of Two Coincident Lines**

Coincident lines are those lines that **lie exactly on top of one anothe**r or in other words **one of the lines** is just **multiplied by** some **scale factor**. Here we will show that **elimination is not applicable** to **any variable** of coincident lines. Suppose two coincident lines below:

x+y=4 [Eq. 1]

2x+2y=8 [Eq. 2]

We can see **equation 2** is **scaled two times** that **of equation 1** making it **coincident.**

To** make** the **first term** of **equation 1** **equal to the first term** of **equation 2,** we will **multiply equation 1 with 2,** which will result in the following:

2x+2y=8 [Eq. 3]

So **in order to eliminate** either the **x or y variable,** we will **subtract equation 2 and equation 3,** which will give us the following:

2x+2y-2x-2y=8-8

0=0

Which will give **infinite solutions.** So in order **to avoid hassle,** we **always** first **check** the **type of equation** then we will **perform the elimination** because we see the** two cases** where **elimination is not permissible.**

**Solved Examples of Systems of Equations Using Elimination**

**Example 1**

Consider the **two sets of equations** first **eliminate the variable x** and then **eliminate the variable y. **Equation is given as:

3x+5y=7 and 2x+3y=5

**Solution**

3x+5y=7 [Eq. 1]

2x+3y=5 [Eq. 2]

**Eliminating Y**

**Multiplying** **equation 1** **by constant 2** and **equation 2** by **constant 3:**

2(3x+5y)=2(7)

6x+10y=14 [Eq. 3]

3(2x+3y)=3(5)

6x+9y=15 [Eq. 4]

Now** both** the **equations** have the c**oefficient of the first term equal,** so will **subtract both equations** that will give us:

6x+10y-6x-9y=14-15

10y-9y=-1

y=-1

y=-1

**Eliminating X:**

3x+5y=7 [Eq. 1]

2x+3y=5 [Eq. 2]

In order **to eliminate y** we will **make the coefficient of y term** of both the equation **equal,** so for that purpose, we will **multiply equation 1** **with 3** and **equation 2 with 5:**

3(3x+5y)=3(7)

9x+15y=21 [Eq. 3]

5(2x+3y)= 5(5)

10x+15y=25 [Eq. 4]

**Subtracting** equations **3 and 4** gives:

9x+15y-10x-15y=21-25

9x-10x=-4

-x=-4

x=4

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