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When Lines Collide
Solving Systems of Linear Equations

 
 If we have two equation lines with different slopes,
 the cross ... SOMEWHERE.
 
 For example, the equations:
 

Y = 3X - 5     and     Y = X + 1
 
 Solving these for X = 0, and X = 2 we get ...
 
(X = 0) Y = 3(0) - 5 Y = (0) + 1
Y = -5 Y = 1
       
X = 0, Y = -5 X = 0, Y = 1
       
(X = 2) Y = 3(2) - 5 Y = (2) + 1
Y = 1 Y = 3
       
X = 2, Y = 1 X = 2, Y = 3
 
 For Y = 3X - 5, two points on the equation line
 are X = 0, Y = -5 and X = 2, Y = 1.
 
 For Y = X + 1, two points on the equation line
 are X = 0, Y = 1 and X = 2, Y = 3.
 
 So we can draw the graphs of these two equations ...
 

 
 We can see from the picture that the two lines do cross.
 If we draw the picture perfectly
 and really, really big,
 we can maybe tell exactly where the lines cross.
 
 But there's a better way ...
 
 At the point where the lines cross
 the X and Y values of the lines are the same numbers.
 

 Well so what?
 What good does that do?
 
 Watch closely ...
 
 If the Y values are the same for both equations,
 we can link the two equations on the same Y.
 

Y = 3X - 5     and     Y = X + 1

3X - 5 =  Y = X + 1
 
 Actually, we don't even need the Y right now,
 so we can get rid of it ...
 

3X - 5 = X + 1
 
 We've solved stuff like this before ...
 
 STEP 1
 Get all of the X's on one side
 
 Here that means subtracting an X from each side ...
 

 
 STEP 2:
 Get all of the numbers on the other side
 
 Here, that means adding 5 to each side ...
 

 
 STEP 3
 Peel the number (the coefficient) away from the X
 
 Here that means dividing both sides by 2.
 

 
 The point where the two lines cross 
 is the point where X = 3.
 
 Now that we have the value of X,
 we can go back to either equation 
 and get the value for Y.
 
 Or do them both for fun.
 FOR FUN????
 
(X = 3)
Y = 3X - 5 Y = X + 1
Y = 3(3) - 5 Y = 3 + 1
Y = 9 - 5 Y = 4
Y = 4
 
 So the place where the two equation lines cross
 is exactly the point: X = 3, Y = 4.
 
 We sometimes write that point (3,4)
 

 
 This point is called the solution "set" of the equations.
 
 That means it is the one and only point
 that is part of both of the lines.
 
 Which means it is the point where the two lines cross.
 
 When we did the calculation, 
 we joined the two equations on the Y.
 We COULD have joined them on the X instead.
 Do it whichever way is easier for you.
 You get the same answer either way.
 

   copyright 2005 Bruce Kirkpatrick
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