Example 1
The population of city A is three times larger than the population of city B. Adding the population of the two cities, we have a total of 200,000 inhabitants. What is the population of city A?
We will indicate the population of cities by an unknown (letter that will represent an unknown value).
City A = x
City B = y
x = 3y
x + y = 200 000
Replacing x = 3y
x + y = 200 000
3y + y = 200 000
4y = 200 000
y = 200 000/4
y = 50 000
x = 3y, replacing y = 50 000
We have
x = 3 * 50 000
x = 150 000
Population of city A = 150 000 inhabitants
Population of city B = 50 000 inhabitants
Example 2
Claudio used only R$20.00 and R$5.00 bills to make a payment of R$140.00. How many notes of each type did he use, knowing that in total there were 10 notes?
x 20 reais bills and 5 reais bills
Equation of the number of grades: x + y = 10
Equation of quantity and value of notes: 20x + 5y = 140
x + y = 10
20x + 5y = 140
Apply Replacement Method
Isolating x in the 1st equation
x + y = 10
x = 10 - y
Substituting the value of x in the 2nd equation
20x + 5y = 140
20(10 - y) + 5y = 140
200 - 20y + 5y = 140
- 15y = 140 - 200
- 15y = - 60 (multiply by -1)
15y = 60
y = 60/15
y = 4
Replacing y = 4
x = 10 - 4
x = 6
Example 3
In an aquarium there are 8 fish, between small and large. If the small ones were one more, it would be twice the big ones. How many are the little ones? And the big ones?
Small: x
Large: y
x + y = 8
x + 1 = 2y
Isolating x in the 1st equation
x + y = 8
x = 8 - y
Substituting the value of x in the 2nd equation
x + 1 = 2y
(8 - y) + 1 = 2y
8 - y + 1 = 2y
9 = 2y + y
9 = 3y
3y = 9
y = 9/3
y = 3
Replacing y = 3
x = 8 - 3
x = 5
Small fish: 5
Big fish: 3
Example 4
Find out which are the two numbers where double the largest plus triple the smallest gives 16, and the largest one plus five times the smallest gives 1.
Major: x
Minor: y
2x + 3y = 16
x + 5y = 1
Isolating x in the 2nd equation
x + 5y = 1
x = 1 - 5y
Substituting the value of x in the 1st equation
2(1 - 5y) + 3y = 16
2 – 10y + 3y = 16
- 7y = 16 - 2
- 7y = 14 (multiply by -1)
7y = - 14
y = -14/7
y = - 2
Replacing y = - 2
x = 1 - 5 (-2)
x = 1 + 10
x = 11
The numbers are 11 and -2.
by Mark Noah
Graduated in Mathematics
Brazil School Team
Equation - Math - Brazil School
Source: Brazil School - https://brasilescola.uol.com.br/matematica/resolucao-problemas-com-sistemas-equacoes.htm