Intersection of 2 Lines

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There are all sort of ways to solve for the intersection of 2 lines. The one I'm going to show you is called Cramer's Rule. This method gets a big much if you are solving larger systems of equations (like 3 equations with 3 unknowns), but it is perfect for solving 2 equations in 2 unknowns.

Before we begin, I really want to stress that you need to know how to add, subtract, multiply and divide integers. Otherwise, it'd be like trying to run the Boston Marathon wearing galoshes (What do you mean you don't know what galoshes are? Look in the dictionary. Or ask your teacher).

As you know, there are 3 possible outcomes as far as a solution is concerned: a) they are the same line; b) the 2 lines are parallel; c) they meet. The beauty of Cramer's Rule is that it tells you right off the bat which case you have.

There are 2 things you have to know about to use Cramer's Rule: 1) what a matrix is, and 2) what a determinant is.

A matrix is just a spreadsheet of numbers that looks like this:

[ 2
4
6
8
]

Just like a spreadsheet, you refer to a matrix by rows and columns. For example, 2 and 6 are in the 1st row, 6 and 8 are in the 2nd column. The dimensions of this matrix is 2 rows by 2 columns, or just 2x2.

If you have a 2x2 matrix of the form:

[ a
c
b
d
]

the determinant is defined as ad - bc.
So, for our matrix above, the determinant would be 2x8 - 6x4 = -8

OK, now we can solve two equations in two unknowns.

Example 1: Find the intersection of -5x + 5y = 8 and -2x - 4y = 6

First, we take the coefficients of x and y and put them in a matrix:

[ -5
-2
5
-4
]

You will notice that the 1st row is from the 1st equation and likewise for the 2nd row. The 1st column numbers are the x coefficients and the 2nd column numbers are the y coefficients.

Now, we find the determinant of this matrix: (-5)(-4) - (5)(-2) = 20 + 10 = 30. We will be using this number to solve for x and y.

To solve for x, what we do is replace the 1st column (the one representing the x coefficients) with the 8 and 6 from the right hand sides of the original equations:

[ 8
6
5
-4
]

and find the determinant of this matrix: (8)(-4) - (5)(6) = -32 - 30 = -62. To solve for x, we divide -62 by 30 and come up with the solution x = -31/15.

To solve for y, we repeat the process, this time replacing the 2nd column with 8 and 6:

[ -5
-2
8
6
]

and find the determinant of this matrix: (-5)(6) - (8)(-2) = -30 + 16 = -14. Just like the way we solved for x, we solve for y by dividing -14 by 30 and we get the solution y = -7/15.

That takes care of the case where the lines intersect. What about the other two cases where either the lines are parallel or they are the same line? Here is how Cramer's Rule works in these cases:
If the determinants of all 3 matrices are zero, both lines are the same line.
If the determinant of the 1st matrix (the number you divide by) is 0, and the determinants of the other two matrices are not 0, you have no solution, which means the lines are parallel.

By the way, if you are lucky enough to take algebra in college or university, you will learn a lot about determinants and how useful they are.

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