Solving systems of linear equations involves finding the values of variables that coordinate with all the given equations on where they will meet. The process typically follows several steps, and there are different methods to achieve a solution. One common method is the substitution method, where one equation is solved for one variable, and then that expression is substituted into the other equations. Another method is the elimination method, which involves manipulating the equations to eliminate one variable by adding or subtracting equations. The goal is to reduce the system to a single equation with one variable, making it easier to find the solution. The purpose of solving systems of linear equations is to determine the points of intersection, if any, between the corresponding lines or planes represented by the equations. These points of intersection, known as the solution set, represent the values of the variables that work with all the equations. The system can have a unique solution, no solution (inconsistent system), or infinitely many solutions (dependent system). The process of solving these systems provides a very good tool for analyzing and understanding complex systems.
Example of Subsitution
4x+2y=17 and x-3y=6 is taken and turned into x=3y+6
You plug that into the other equation.
4(3y+6)+2y=17 -> 12y+24+2y=17 -> subtract 24 on each side and add the y’s -> 14y=-17=-0.5
Y=-0.5
X= 4x+2(-0.5)=17 -> 4x+1=17 subtract one on both sides 4x=16 -> 4x/4=16/4=4 X=4
(4,-0.5)
Example of Elimination
4x+2y=8 and 2x+y=2 -> 2(2x+2=2) now the new system is
4x+2y=8 and 4x+2y=4
The 4xs cancel out when you add it the 2y and 2y add up and 4 and 8. Making it 4y=8 -> 4y/4=8/4 Y=4
Then plug it in
4x+2(4)=8
4x+8=8 subtract 8 from both sides
4x=0 -> 4x/4=0/4 X=0
(0,4)