Systemslinear they are sets in equations in which the incognitos have the same value regardless of the equation they are in. O method gives replacement is one of the options available to solve this type of problem.
for a set in equations be considered a system, it is necessary that incognitos equals represent equal numbers. In this case, we use “open curly” (the symbol { is open curly) to represent this relationship between the equations. So, it's an example of a system:
Looking at the equations separately, x = 2 and y = 1 is a possible result. Check this by putting 2 for x and 1 for y and doing the math. To system, this is the only possible outcome.
solve one system, therefore, is to find the x and y values that make it true.
Replacement method
This method basically consists of three steps:
Find the algebraic value of one of the incognitos using one of the equations;
To replace this value in the other equation. With that, the numerical value of one of the unknowns is found;
To replace the numerical value already found in one of the equations to discover the value of the unknown unknown.
As an example, look at the following solution of a system:
For the first step, we can choose any of the equations. We always suggest choosing the one that has at least one unknown with coefficient 1 and this must be the unknown that will have its algebraic value found. We will, therefore, choose the second one and find the algebraic value of x. This procedure is also known as “isolateTheunknown”, so we can also say that we will isolate x:
x + y = 20
x = 20 - y
Note that for this process we only use the rules for solving equations.
The second step is to replace the value of this unknown at other equation. Note that it is not allowed. to replace the value of x in the same equation already used. Thus, we will have:
5x + 2y = 70
5·(20 - y) + 2y = 70
applying to distributive property:
100 - 5y + 2y = 70
– 5y + 2y = 70 – 100
– 3y = – 30
3y = 30
y = 30
3
y = 10
To complete the third step, just to replace the value of unknown found in any of the equations. We will choose the second one because it has the smallest coefficients.
x + y = 20
x + 10 = 20
x = 20 - 10
x = 10
The solution of system above is x = 10 and y = 10, which can also be written as follows: S = {10, 10}. If the latter is used, be sure to enter the x value first and then the y value: S = {x, y}.
By Luiz Paulo Moreira
Graduated in Mathematics
Take the opportunity to check out our video lesson on the subject:
