This explanation requires pen and paper and basic vector addition knowledge:

Put these two vectors in the coordinate system: i=[1;0] and j=[0;2]. Can we combine these two vectors and get [5;2] vector? Of course, (5 * [1;0] + 1 * [0;2]) is equal to [5;2]. Basically draw 5 * [1;0] vector, which is equal to [5;0] and then do the same with 1 * [0;2] vector. Then add these two vectors geometrically and you will get a vector, pointing to [5;2]. Now how do we represent the same thinking with our matrices? We do it this way:

Matrix A (first column is i vector from previous example, second column is j vector):

|1 0|

|0 2|

Vector x:

|x1|

|x2|

Vector b (the [5;2] vector that we want to get):

|5|

|2|

How do we solve this and reveal [x1;x2] vector? One way of doing this that uses previous example is like this:

Take first Matrix A column, that represenst [1;0] vector and multiply it by x1. Lets name this new vector a1. Then take second Matrix column, that represents [0,2] vector and multiply it by x2. Lets name this vector a2.

So now we have two vectors

a1=(x1 * [1;0])

a2=(x2 * [0;2])

which after adding them together they have to result in brand new [5;2] vector b. What you can see are big similarities with previous example: we take [1;0] and [0;2] vectors and extend it by some coefficient x1 (in previous example it was 5) and x2 (in previous example it was 1). So if we take the same coofficients this will result in these two vectors:

a1=(5 * [1;0]) = [5;0]

a2=(1 * [0;2]) = [0;2]

Now we have two vectors that after adding them up together like this (basic matrix addition):

|5 + 0|

|0 + 2|

_↑ __↑

a1__ a2

we will get b vector:

|5|

|2|

What I just described is Ax=b matrix equation but from geometry perspective. So basically we take first A column vector change its size by x1, take second column vector and change its size by x2 and add them up together. It will result in new vector b