In this post we will learn about the concept of simultaneous linear equation and some methods to solve them.

At the end of the chapter, some solved examples are also provided for conceptual clarity.

## What are simultaneous linear equation ?

The **set of two linear equations** containing** two or more variables** are called **simultaneous equation**.

These **equations contain common solutions** which will satisfy both the equations.

Let us understand the concept with an example.

**Consider the below two linear equation.**

3x + 2y = 5

-x + y = 0

Both the linear equation contain two variables x and y.

On solving, **you will get value of x & y which will satisfy both the equation**.

Here **common solution of x & y is ( 1, 1 )**.

Let’s check if the value (1, 1) satisfy both the given equation.

**First Equation**

3x + 2y = 5

Putting value (1, 1) in the equation.

3(1) + 2(1) = 5

3 + 2 = 5

5 = 5

LHS = RHS

Hence **for (1,1), the equation is satisfied.**

**Second equation**

-x + y = 0

Putting (1 ,1) in the equation.

-1 + 1 = 0

0 = 0

LHS = RHS

Hence **for (1, 1), the equation is satisfied.**

From the above calculation, one can conclude that **(1,1) is a common point which satisfy both the equation**.**In graphical form**, the above two linear equation can be expressed as;

Equation **3x+2y = 5** is shown by **orange line**.

Equation **-x + y = 0** is shown by **black line**.

The solution point (1, 1) is at the** intersection of both the lines**. It means that point (1, 1) **lies at both the lines**.

**Conclusion**

Below are features of simultaneous equation.

(a) It contains two linear equations.

(b) Each linear equation contains two or more variables.

(c) The equations have at least one common solution

## How to solve simultaneous equations ?

Here we will discuss three methods to solve simultaneous equations.

(a) Substitute Method

(b) Elimination method

(c) Graphical method.

We will understand each of the method using solve example.

### Substitute Method to solve simultaneous equation

If x & y are two variables in given set of linear equations then simultaneous equation can be solved **using following steps**;

(a) **Find value of x with respect to y**.

(b) **Now substitute value of x **in the 2nd equation.

After solving the second equation, you will find exact value of y.

(c) **Now use this value of y to find exact value of x**.

I hope you understand the above process. Given below is the solve example for further clarity.

**Example 01**

Solve the below simultaneous equation using substitute method.

x + 3y = 6

2x – 3y = 12**Solution**

Taking first equation.

x + 3y = 6**Find the value of x with respect to y.**

x = 6 – 3y**Now put this value of x in second equation**.

2x – 3y = 12

2 (6 – 3y) – 3y = 12

12 – 18y – 3y = 12

-21y = 12 – 12

-21y = 0

y = 0

Here we got the exact value of y.

Now put the value of y in any of the two equation and you will get the value of x.**Let’s take the first equation again**.

x + 3y = 6

Put y = 0;

x + 3(0) = 6

x = 6

Hence, **the point (6,0) is the solution** of given simultaneous equation.

**Example 02**

Solve the below equation using elimination method.

2x – y = -1

3x+ 2y = 9

**Solution**

Take the first equation and **get the value of y with respect to x**.

2x – y = -1

– y = -1 -2x

-y = – (1+2x)

y = 1+2x**Now put this value of y in second equation**.

3x + 2y = 9

3x + 2 (1+2x) = 9

3x + 2 + 4x = 9

7x = 7

x = 7 / 7

x = 1

Here we got the exact value of x.

Now put this x value in any of the two equation and you will get exact value of y.**Taking first equation**.

2x – y = -1

2(1) – y = -1

2 – y = -1

y = 2 + 1

y = 3

Hence, **(1,3) is the solution** of given simultaneous equation.

### Elimination method to solve simultaneous equation

To solve the simultaneous equation, **follow the below steps**;

(a) **Vertically arrange the equation** such that** same variables are at same line.**

(b) **Multiply/Divide one of the equation** to make coefficient of one variable equal to each other.

(c) **Now subtract/add the equation to cancel out the common variable **and get the exact value of other variable.

Using the above three steps you can solve the given simultaneous equation. Given below are solved examples for further clarity.

**Example 01**

Solve the below equations;

2x + 5y = 20

6x – 5y = 12

**Solution**

Vertically arrange the equation such that same variables are in same line.

Note that **both expressions have same coefficient of y**. So when we do vertically addition, variable y will get eliminated and we will get exact value of x.

Further solving the expression.

8x = 32

x = 32 / 8

x = 4**Hence, we got the exact value of x**.

Now put this value of x in any of the above two equation and you will get value of y.

Again taking first equation.

2x +5y = 20

2 (4) + 5y = 20

8 + 5y = 20

5y = 20 – 8

5y = 12

y = 12 / 5

y = 2.5

Hence,** (4, 2.5) is the solution of given equation.**

**Example 02**

Solve the below equation using elimination method.

5x + 4y = 22

7x + 6y = 32

**Solution**

Arranging both the expression vertically.

To make the x coefficient same in both the equation, multiply first equation by 7 and second equation by 5.

After multiplication we get;

Note that we have linear equation with **same x coefficient**. **To cancel the x variable, subtract both the equations**.

On subtracting the equations we get;

-2y = -6

y = 6 / 2

y = 3

Here we got exact value of y. Put this value of y in main equation to get value of x.

Selecting first equation;

3x + 4y = 22

3x + 4 (3)= 22

3x + 12 = 22

3x = 10

x = 10/3

Hence,** (10/3, 3) is the solution of given equation**.

### Graphical Method

To solve simultaneous equation using graphical method, follow the below steps;

(a) **Plot the graph of both the equation** independently.

(b) The** point of intersection of two equation is the solution**.

I hope you understood the above process. Let us see below example for further clarity.

**Example 01**

Solve the below simultaneous equation using graphical method.

2x + 5y = 6

3x + y =11

**Solution**

Plotting the graph of both the linear equation.

Note that both the lines intersect at point (3.7, -0.3).

Hence **(3.7, -0.3) is the solution** of given simultaneous equation.