# How do i solve a system of 2 equations?

This problem is one that can be confusing if not tackled orderly. Let us have an example of a system of 2 equation:

**3x + 5y = 13**

**4x + 7y = 18**

First we must choose one of the two and then express one of our variables with respect to the other. Let's take the first one and express **y **with respect to **x**:

**3x + 5y = 13**

**5y = 13 - 3x**

**y = 13/5 - 3x/5**

Already it doesn't look very nice, but do not panic! the next step is to use this formula for the next equation. By substituting **y** in the second equation with this function of **x** we will have a single equation with only **x** as an unknown:

**4x + 7y = 18**

**4x + 7*(13/5 - 3x/5) = 18**

Now we group the **x** factors toghether:

**4x + 7*13/5 - 7*3 x/5 =18**

**(4 - 7*3/5)*x + 7*13/5 =18**

We put the constants on the right hand side and divide the whole equation by the **x** factor.

**(4 - 21/5)*x = 18 - 7*13/5**

**x = (18 - 91/5) / (4 -21/5)**

Now it all comes down to a simple calculation for the **numerator **and **denominator**.

**numerator = (18*5-91)/5 = -1/5**

**denominator = (4*5-21)/5 = -1/5**

In this case they are equal, so we find **x**

**x = 1**

Now we substiture this result in the first equation and we can find the last unknown, **y**

**3x + 5y = 13**

**3*1 +5y = 13**

**5y = 13-3**

**5y = 10**

**y = 2**

The answers are **x=1 **and **y****=2**. This is the simplest way to solve the 2 equation system and it work for any system. There are some methods that can solve system of more than 2 equations, but that is a topic for another time.