Linear Equations
Patterns within systems of Linear Equations
HL Type 1 Maths Coursework
Maryam Allana
12 Brook
The aim of my report is to discover and examine the patterns found within the constants of the linear equations supplied. After acquiring the patterns I will solve the equations and graph the solutions to establish my analysis. Said analysis will further be reiterated through the creation of numerous similar systems, with certain patterns, which will aid in finding a conjecture. The hypothesis will be proven through the use of a common formula. (This outline will be used to solve both, Part A and B of the coursework)
Part A:
Equation 1: x+2y= 3
Equation 2: 2x-y=4
Equation 1 consists of three constants; 1, 2 and 3. These constants follow an arithmetic progression with the first term as well as the common difference both equaling to one. Another pattern present within Equation 1 is the linear formation. This can be seen as the equation is able to transformed into the formula ‘y = mx+c’ as it is able to form a straight line equation (shown below). Similar to Equation 1, Equation 2 also follows an arithmetic progression with constants of; 2, -1 and 4. It consists of a starting term of 2 and common difference of -3. As with Equation 1, Equation 2 is also linear forming the formula ‘y = mx+c’. When examining both Equation 1 and 2, an inverse pattern can be seen, where equation 1 is the inverse of equation 2 and vice versa. This can be proved by observing gradients of both the equations where equation 1 equals ‘y= -x/2 + 3/2’ and equation 2 equals ‘y= 2x+4’ (This is proven through technological means below)
x + 2y= 3- Equation 1
2x - y= 4- Equation 2
The equation must be solved simultaneously in order to acquire a solution. Therefore…
x + 2y= 3
4x - 2y= -8
-----------------
5x= -5/5 thus x= -1 and y= 2
Graph
The significance of the solution is the point of intersection with values of x= -1 and y=2 as seen with both the
HL Type 1 Maths Coursework
Maryam Allana
12 Brook
The aim of my report is to discover and examine the patterns found within the constants of the linear equations supplied. After acquiring the patterns I will solve the equations and graph the solutions to establish my analysis. Said analysis will further be reiterated through the creation of numerous similar systems, with certain patterns, which will aid in finding a conjecture. The hypothesis will be proven through the use of a common formula. (This outline will be used to solve both, Part A and B of the coursework)
Part A:
Equation 1: x+2y= 3
Equation 2: 2x-y=4
Equation 1 consists of three constants; 1, 2 and 3. These constants follow an arithmetic progression with the first term as well as the common difference both equaling to one. Another pattern present within Equation 1 is the linear formation. This can be seen as the equation is able to transformed into the formula ‘y = mx+c’ as it is able to form a straight line equation (shown below). Similar to Equation 1, Equation 2 also follows an arithmetic progression with constants of; 2, -1 and 4. It consists of a starting term of 2 and common difference of -3. As with Equation 1, Equation 2 is also linear forming the formula ‘y = mx+c’. When examining both Equation 1 and 2, an inverse pattern can be seen, where equation 1 is the inverse of equation 2 and vice versa. This can be proved by observing gradients of both the equations where equation 1 equals ‘y= -x/2 + 3/2’ and equation 2 equals ‘y= 2x+4’ (This is proven through technological means below)
x + 2y= 3- Equation 1
2x - y= 4- Equation 2
The equation must be solved simultaneously in order to acquire a solution. Therefore…
x + 2y= 3
4x - 2y= -8
-----------------
5x= -5/5 thus x= -1 and y= 2
Graph
The significance of the solution is the point of intersection with values of x= -1 and y=2 as seen with both the