it cost 0.45 to send to europe and 0.65 to send to africa, and a total of 18.55 was spent. so the cost of letters to europe is 0.45 times however many letters went there, which we'll call 'e'. and the cost of the letters to africa is 0.65 times however many letters went to africa, which we'll call 'a' the sum or total of both of those amounts is 18.55. so

(0.45e) + (0.65a) = 18.55

we also know that the total number of letters sent to europe and africa is 35. since we used 'e' to represent the total number of letters to europe, and 'a' to represent the total number of letters to africa, we're talking about the same variables as above, so

e + a = 35

now we have two equations with two variables. we need to solve one of the equations for one of the variables, then use that in the other equation to find the value of the other variable. once we know the value of one variable, we can plug that amount back into either equation to solve for the remaining variable. may sound confusing, but stay with me. let's solve one of the equations for one of the variables.

e + a = 35

subtract a from both sides

e = 35 - a

so wherever we have e, we can just substitute in 35 - a (we'll be sure to put it in parenthesis so we know it goes together). if we plug that value in for e in the same equation, it will just cancel everything out, so we'll plug it in for e in the other equation.

(0.45e) + (0.65a) = 18.55

now plug it in to get

(0.45*(35-a)) + (0.65a) = 18.55

now multiply 0.45 times everything inside the parenthesis next to it to get

15.75 - 0.45a + 0.65a = 18.55

now let's get the numbers with the variable on one side, and the numbers without the variable on the other by subtracting 15.75 from both sides:

-0.45a + 0.65a = 2.8

now, since everything on the left side has the same variable, we can pull that out like this

a(-0.45+0.65) = 2.8

simplify

a(0.2) = 2.8

now divide both sides by 0.2 to get

a = 14

now we know what the value of 'a' is, so we can plug it back into either equation, and it should work. let's do the simple one.

e + a = 35

turns into

e + 14 = 35

subtract 14 from both sides to get

e = 21

now, to check our work, let's plug both e and a back into the other equation. if we're right, it should all come out equal.

(0.45e) + (0.65a) = 18.55

plug in both values

(0.45*21) + (0.65*14) = 18.55

simplify

9.45 + 9.1 = 18.55

simplify again

18.55 = 18.55

yep, that checks out, so we know we found the right answer. there were 21 letters to europe and 14 letters to africa.

*phew* :)



to review, here are the steps we took.
1) write down the equations using variables for each thing we don't know (in this case, two equations and two variables)

2) solve one of the equations for one of the variables

3) plug the result (the other side of the equals sign from the variable we solved for) back into the other equation wherever that variable is.

4) now solve for the one remaining variable to find its actual value

5) plug this actual value into one of the equations to solve for the other variable

6) (optional) plug both values into the other equation to make sure it still works. it should, but this is just a nice check. if it doesn't, you messed up somewhere

this same process can be used to solve for three variables, though you'll (probably) need three equations. when you solve for one variable, then you'll be down to two, then you'll solve for one variable again, so it just adds a layer of complexity. this can go on for however many variables you may be given!