2 Answers

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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