Respuesta :
Answer:
Step-by-step explanation:
You have two equations given and you want to solve all the couples (x; y) ∈ [tex]R^{2}[/tex] that verifies both equations at the same time.
[tex]S:\left \{ {{2x-3y=5} \atop {-4x+5y=7}} \right.[/tex]
First Method:
Substitution:
you need to give an expression of y in function of x or x in function of y in one of the two equations of the linear system, in order to replace the obtained expression in the other equation.
Example:
Take 2x-3y=5 <==> 2x = 5+3y <==> x = 5/2 + 3/2y
And now replace the obtained x in the other equation:
-4x + 5y = 7 <==> -4*(5/2+ 3/2*y) + 5y = 7 <==> -10 - 6y + 5y = 7
<==> -y = 17 <==> y = -17
and then you need to obtain x: 2x-3y=5
2x - 3*(-17) = 5 <==> 2x + 46= 0 <==> x = -23.
And the solutions are S = {(-23; -17)} ∈ [tex]R^{2}[/tex]
Second Method:
Linear Combination:
You combine the two equations in order to have only one unknown value or variable an then you solve the set.
| 2x - 3y = 5 (*2)
| -4x + 5y = 7
| 4x -6y = 10
| -4x + 5y = 7
| 0 - y = 17 <==> y = -17
and then you need to obtain x: 2x-3y=5
2x - 3*(-17) = 5 <==> 2x + 46= 0 <==> x = -23.
So: S = {(-23; -17)} ∈ [tex]R^{2}[/tex]
