Systems of Linear Equations. Investigate Systems of linear equations where the system constants have well known mathematical patterns.

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Mathematical Investigation

Investigate Systems of linear equations where the system constants have well known mathematical patterns.

Part A:

Here is a 2x2 system of linear equations:            

                                                 

It is easy to notice that if one starts from the coefficient of x, they can generate the coefficient of y by adding 1 to it and the constant by adding 1 to the coefficient of y. Another way to look at this is to start from the constant and add -1 to it you can generate the y coefficient and add -1 to this to get the x coefficient. The same is true for the second equation except that instead of adding 1 to the coefficient of x, we add -3. Similarly, we can start from the constant and add 3 each time to get the y and x coefficients.

       x + 2y= 3           x= -2y + 3                                                                      x= -2(2) + 3        x= -1

       2x – y= -4         2(-2y + 3) – y= -4        -4y + 6 – y= -4       -5y= -10          y= 2                    y= 2

The solution x equals -1 and y equals 2 means that that point satisfies both equations. As we can see in the graph the two equations intersect at a point and that point is the same as the solution we found when we solved the problem algebraically. So the point that the two equations intersect is (-1, 2).

         Graph

Here are some examples using the same pattern:

Example 1:

x + 3y= 5        x= 5 – 3y                                                                         x= 5 – 3(2)       x= -1 

5x – y= -7      5(5 – 3y) – y= -7      25 – 16y= -7       -16y= -32         y=2                    y= 2 

Example 2:

2x + 6y= 10         x= 5 – 3y                                                         x= 5 – 3(2 )      x= -1

Join now!

7x + 3y= -1       7(5 – 3y) + 3y= -1       35 – 18y= -1        -18y= -36       y= 2                    y= 2

In the first example, the pattern in the coefficients for the first equation was adding two. In the second equation the pattern was subtracting six or adding negative six. The solution to the first example was the same as the solution in the original system of equations. In my second example, the pattern in the coefficients for the ...

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