Equivalent systems- Systems of equations that have the same solution.
Linear Systems was mostly founded by Carl Friedrich Gauss. He was a German scientist and mathematician who excelled in mathematics. Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians. Gauss earned a scholarship and in college, he independently rediscovered several important theorems; his breakthrough occurred in when he was able to show that any regular , each of whose odd factors are distinct , can be constructed by alone, thereby adding to work started by classical Greek mathematicians. Unfortunately for mathematics, Gauss reworked and improved papers incessantly, therefore publishing only a fraction of his work. But in the end he succeeded throughout his life and he was one of the most memorable mathematicians throughout history.
Laws
The graphs of two linear equations in two variables may intersect at one point, be parallel and distinct, or coincide.
Theorems
Substitution Method
With the substitution method, we solve one of the equations for one variable in terms of the other, and then substitute that into the other equation.
Solve for x in equation (1)
Substitute (3-2y) for x in equation (2)
Substitute 1 for y in equation (1)
Therefore the solution is (1, 1).
Theorems
Elimination Method
In the ‘elimination’ method for solving simultaneous equations, two equations are simplified by adding them or subtracting them. This eliminates one of the variables so that the other variable can be found.
2x - 5y = 1
3x + 5y = 14
Add the equations eliminates the y terms and will create an equation an equation with one variable
5x = 15
x = 3
Substitute 3 for x in equation (1)
2x - 5y= 1
2(3) – 5y= 1
6 – 5y= 1
6 – 1= 5y
5= 5y
y= 1
Therefore the solution is (3, 1).
