Simultaneous equations are a set of equations with multiple variables, such as x and y, a and b, etc. that need to be solved at the same time i.e. simultaneously. By solving these equations, we can determine the values of the variables that satisfy all the given equations. This process allows us to establish a relationship between the variables and find their specific values. Solving simultaneous equations is a fundamental tool in **IGCSE Mathematics** and is used to solve various real-world problems that involve multiple unknowns.

## Techniques To Solve Simultaneous Equation

There are different **methods to solve simultaneous equations**, including elimination and substitution. In this blog we will be covering the substitution method however, **BOTH these methods are crucial for exams.**

**SUBSTITUTION METHOD**

The substitution method involves solving one equation for one variable and substituting it into the other equation. It is generally used when you have expressions where it is easy to rearrange the given expression in the form of **y = ?** or **x = ?**

__Step 1:__ **Write down the given equations.**

Equation 1: **3***x*** + y = 19**** **

Equation 2: **x + y = 9**

__Step 2:__ **Solve one equation for one variable.**

Let's solve Equation 2 for **y**:

**x + y = 9 **

**y = 9 – x ** (This has been done by following simple algebraic rules)

__Step 3__: **Substitute the expression for y into Equation 1:**

Now we consider the first equation (**3x + y = 19**), and instead of the **y**, we replace it with the expression that has been rearranged (**y = 9 – x**). By doing that, we can solve and find the value for **x**, because now** **** y** has been removed from the equation.

It is now a simple linear equation.

**3x + ****(9 – x)**** ****= 19**** **

__Step 4:__ **Now, solve the equation to find the value of x:**

**3x + 9 – x = 19**

**2x = 19 – 9 **

**2x = 10 **

**2x = 10** (both sides need to be divided by 2)

** **__x = 5__

This is **NOT** the end of your work. Since you have found the value of x, you must now use it to find the value of y.

We can consider the second equation we rearranged** (y = 9 – x)**, and here, instead of** x**, we are going to substitute it with the value of **x** that we found earlier.

**y = 9 – x**

As** ****x = 5****, ***y ***= 9 (–**** ****5)**

** **__y = 4__

This way, we have found what both** x** and** y** stand for. In any simultaneous equation, you must provide an answer for both the unknown variables, both x and y, or a and b, or any two letters which are involved.

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