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Line Intersections & Systems of Equations
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Parallel and perpendicular lines, intersection of 2 straight lines, solving a two-variable system of equations
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Parallel lines
SL Core 2.1
Two lines are parallel when they have the same gradient m and they do not intersect:
m1x+c1∥m2x+c2⇔m1=m2 and c1=c2
In this case, the system of equations formed by the two lines has no solutions.
If the lines have the same gradient and they intersect, then they must be the same line.
Intersections of straight lines
SL Core 2.1
Suppose we have the straight lines y=3x−2 and y=2−3x. Where do the lines intersect?
Lines intersect when they have a point in common. That is, for some x:
3x−2=2−3x
Rearranging gives
6x=4⇒x=32
If two lines do not intersect, then they must be parallel, since the definition of parallel is two straight lines that never meet.
If two lines are the same (possibly in different forms), then their intersection will all real numbers.
Perpendicular Lines
SL Core 2.1
Two lines are perpendicular if they form a right angle with respect to each other. In this case, the rise of one line becomes the run of the other, with a sign change:
m1×m2=−1
Systems of equations with 2 unknowns
SL Core 2.1
We can solve a system of 2 equations and 2 unknowns with different methods.
{3y+2x−2=03y−3x+1=0
By substitution
Rearranging
3y+2x−2=0⇒y=−32x+32
Substituting this into 3y−3x+1=0:
(−2x+2)−3x+1=0
−5x=−3
So x=53, which implies y=−32⋅53+32=154. So the intersection is (53,154).
By elimination
We can eliminate y from the equations by subtracting the second from the first:
(3y+2x−2)−(2y−3x+1)2x−2+3x−15x=0=0=3
So x=53⇒y=154 and the intersection is again (53,154).
We can use either of these methods to systems of equations with 2 equations and 2 unknowns.
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