Before we start talking about *simultaneous equations*, let’s shed light on what equations in maths are and their types. In Mathematics, an equation is a mathematical statement in which two values should be equal. Furthermore, it consists of two expressions on each side of an equal sign (=). Consists of two or more variables. In short, your L.H.S value should be similar to the R.H.S value. While substituting values of the variables in an equation, they should be equal. Besides, there are different types of equations in Maths, like:

- Linear Equation
- Quadratic Equation
- Polynomial Equations

Notably, many students need help with tasks **solving simultaneous equations**. More practice is required for them to get the correct answer consistently. So if you are also looking for a helping hand then, in this article, we will discuss **How to Solve Simultaneous Equations**. Now, let’s try to learn what are simultaneous equations.

## What Are Simultaneous Equations?

The simultaneous equation involves two or more quantities related using two or more equations. Moreover, it includes a set of few independent equations. Simultaneous equations, also known as the system of equations, in which it holds a finite set of equations for which the standard solution is sought. However, we need to find the values of the variables included in these equations to solve simultaneous equations questions. They are also called simultaneous equations because equations are translated at the same time.

**For Example:**

2x + 4y = 14 4x − 4y = 4 6a + b = 18

4a + b = 14 3h + 2i = 8 2h + 5i = −2

Each of these equations on their own has infinite possible solutions.

## Way For Solving Simultaneous Equations

We can **solve simultaneous equations** using various methods. Moreover, there are three ways to solve substitution, elimination, and augmented matrix methods. Among these methods, the two are the simplest methods to get accurate solutions. Furthermore, here we are going to discuss these critical methods, such as:

- linear simultaneous equations
- quadratic simultaneous equations

Linear equations contain terms that are raised to a power that is no higher than one.

**For Example**

2x+5=0

*Linear simultaneous equations* are usually solved by the elimination method (although the substitution method is also available for you).

Next, we talk about quadratic equations, which contain terms raised to a power no higher than two.

**For Example**

E.g.

**x2−2x+1=0**

Although the questions of quadratic **simultaneous equations are** solved with the help of the substitution method,

## Solving Simultaneous Linear Equations Using The Elimination Method

Follow the solved example below to understand the method of **solving simultaneous equations** by the elimination method and the steps.

4a + 5b = 12,

3a – 5b = 9

**Solution**

**Solution:**

The two given equations are

4a + 5b = 12 …….(1)

3a – 5b = 9……….(2)

*STEP 1*

The coefficient of variable ‘b’ is equal and has the opposite sign to the other. As well as you can add equations 1 and 2 to eliminate the variable ‘b’ and make a new one.

*STEP 2*

The terms will be

(4a+3a) +(5b – 5b) = 12 + 9

7a = 21

*STEP 3*

Bring the coefficient to the R.H.S of the equation

a = 21/ 7

*STEP 4*

After dividing the R.H. S of the equation, we get a = 3

*STEP 5*

Now, substitute the value a=3 in the equation (1)

4(3) + 5b = 12,

12 + 5b = 12

5b = 12-12

5b =0

b = 0/5 = 0

*STEP 6*

Hence, the solution of the given equations is a = 3 and b = 0.

## Solving Simultaneous Linear Equations With The Substitution Method

Let’s try to solve **simultaneous equation question****s** better using the substitution method.

b= a + 2

a + b = 4.

**Solution**

The two given equations are

b = a + 2 ————–(1)

a + b = 4 ————–(2)

*STEP 1*

Substitute the value of b in the second equation.

a + (a + 2) = 4

*STEP 2*

Solve for a

a +a + 2 = 4

2a + 2 = 4

2a = 4 – 2

a = 2/2 = 1

*STEP 3*

Substitute the value of an in equation 1

b = a + 2

b = 1 + 2

b = 3

*STEP 4*

Hence, the solution for the given simultaneous equations is: a = 1 and b = 3