Grade 10
196298501
Patterns within systems of linear equations
Systems of linear equations are a collection of linear equations that are related by having one solution, no solution or many solutions. A solution is the point of intersection between the two or more lines that are described by the linear equation. Consider the following equations: x + 2y = 3 and 2x – y = 4. These equations are an example of a 2x2 system due to the two unknown variables (x and y) it has. In one of the patterns, by multiplying the coefficient of the y variable by 2 then subtract the coefficient of x from it you will be given the constant. As a word equation it can be written like so with the coefficient of x as A and coefficient of y as B and the constant as C, 2B – Ax = C. This can be applied to the first equation (x + 2y = 3) as 2(2) – 1 = 3. To the second equation (2x – y = 4), it is 1(2) – 2 = 4. By using matrices or graphs, we can solve this system. Regarding other systems that also has such as pattern, it should also have the same solution as the two examples displayed. For instance, 3x + 4y = 5 and x 2y = 5, another system, also displays the same pattern as the first set and has a solution of (1, 2). Essentially, this pattern is indicating an arithmetic progression sequence. Arithmetic progression is described as common difference between sequences of numbers. In a specific sequence, each number accordingly is labelled as an. the subscript n is referring to the term number, for instance the 3rd term is known as a3. The formula, an = a1 + (n – 1) d, can be used to find an, the unknown number in the sequence. The variable d represents the common difference between the numbers in the sequence. In the first equation (x + 2y = 3) given, the common differences between the constants c – B and B – A is 1. Variable A is the coefficient of x and variable b represents the coefficient of y, lastly, c represents the constant. The common difference of the second equation (2x – y = 4) is 3 because each number is decreasing by 3. In order to solve for the values x and y, you could isolate a certain variable in one of the equations and substitute it into the other equation. x + 2y = 3
2x – y = 4
x + 2y = 3
* x = 3 – 2y
* 2(3 – 2y) – y = 4
* 6 – 4y – y = 4
* 6 – 5y = 4
* 5y = 10
* y = 2
Now that the value of y is found, you can substitute 2 in as y in any of the equations to solve for x. x + 2y = 3
* x + 2(2) = 3
* x + 4 = 3
* x = 3 – 4
* x = 1
Solution: (1, 2)
Even though the solution has already been found, there are many different ways to solve it, such as graphically solving it. By graphing the two linear lines, you can interpolate or extrapolate if necessary to find the point where the two lines intersect.        

       
       
       
       
       
       
       
       
       
       
       
     Graph 1
Graph 1
  
       
       
       
       
Just from the equations given, it is not in a format where it can be easily graphed. By changing it into y=mx + b form, the first equation will result as y =  (1/2) x + 3/2 or y = 0.5x + 1.5 and the second equation will result as y = 2x + 4. The significance of the solution is that it is equal to the point of intersection as shown on Graph 1. This can then allow the conclusion that the solution of the two linear equations is also the point of intersection when graphed. According to this arithmetic progression sequence, it could be applied to other similar systems. For instance, the examples below demonstrates how alike 2x2 systems to the previous one will display a similarity. Example 1: In the first...