Okay, lets use x to be the age of the son, and y to be the age of the mother.

Then the first equation says: a mother is 7 times as old as her son so:

y = 7x

Pretty straight forward.

And the second equation says: In five years time she will be 4 times her son's age.

This one takes a little more thought. if she is y years old now, then she will be y+5 years old in 5 years. So the LHS (left hand side) becomes y+5. And at that time she will be 4 times as old as her son, so the RHS becomes 4x. However, her son must have also aged 5 years, so x needs to be replaced with (x+5) and thus 4x becomes 4(x+5)

y+5 = 4(x+5)

Now we have two equations, lets substitute the top equation into the bottom equation:

y+5 = 4(x+5) becomes 7x+5 = 4(x+5)

Now we can distribute the 4 on the RHS

7x+5 = 4x+20

subtract 4x and subtract 5

3x = 15

divide by 3

x = 5

So we know that the son is 5 years old now. Lets substitute this value back into either equation to get a value for y. I'll do the top one since it is a simpler equation:

y=7x

y=7*5

y=35

So the mother is 35 years old, and the son is 5 years old.

To check our work, we can see that she is 7 times as old as he is, and that in 5 years she will be 40 and he will be 10, and at that time she will be 4 times as old as he is. So we know we did it correctly.