Hi:
there are three ways to solve a linear systems of equations:
1. Graph it - very cude but will yield good answers
2) substitution - better than number 1
3 Elimination - Best method for find the answer
Example for substitution (#2) :
Given:
x+ 1 = y
2x+3 = y
x+1 = 2x +3 - substitution
-x + x +1 = 2x+3 + - x - Adding the additive inverse of a variable to both sides of the equation to move it to the other side of it
1= x+3 - Addition
-3+ 1 =x + 3 + -3 - Adding the additive inverse of a number to both sides of the equation to move it to the other side of it
-2 = x - Addition
Let solve for y
x+ 1 = y - one of the given equation from the above
-2 + 1 = y - substituting x with -2
-1 = y - Addition
solution is - 2, -1
Proof:
Given :
x+ 1 = y
2x+3 = y
step 2
-2+ 1 = -1
2(-2)+3 = -1- substituting xwith -2 and y with -1
step 3:
-2+1 = -1
-4+3 = -1 - multiplication of -2 and 2
-1 = - 1
-1 = -1 - Addition
It checks and equals
Example for Elimination (#3) :
Given:
x+ 1 = y
2x+3 = y
Step 1:
I need a common multiple to cancel y to solve for x, -1 is good . So I'll use it
step 2:
(-1) x+ 1 = y( -1)
2x+3 = y - Multiplying a negative number to one of the equation to both sides of it to change a coefficient from a negative value to a positive one or vise versa to cancel one of the terms to solve for the other though the addition of terms
-x-1= - y
2x+3= y
----------- - Multiplication and addition of terms
x +2 = 0
x+ 2 + -2 =0 + -2 - Adding the additive inverse of a number to both sides of the equation to move it to the other side of it
x = -2 - Addition
step 3:
let solve for y :
x+ 1 = y
2x+3 = y
I need a common multiple to cancel x to solve for y, -2 is good . So I'll use it
(-2) x+ 1 = y (-2)
2x+3 = y - Multiplying a negative number to one of the equation to both sides of it to change a coefficient from a negative value to a positive one or vise versa to cancel one of the terms to solve for the other though the addition of terms
-2x-2 = -2y
2x+3 = y
---------------- - Multiplication and addition of terms
1 = - y
(-1)1= (-1)-y - Multiplying a negative number to both sides of the equation to
remove a negative variable to get a positive variable
-1 = y
solution ( -2,-1)
I don't need the do the proof again since I really did it in the above
However you will get one of three answer for a system of equations
1) it as a solution
2) there is no solution :due the fact they are a parrell line ( exp: y = 2x +1 ,y = 2x+3)
3 there are a infinite number of soultions for it: due the fact they are the same line ( example :
2x + 3 = y , 4x+ 6 = 2y )
I hope this helps