Simultaneous Equations Flashcards
“There is a fixed ratio between the variables and constants to all 3 equations”
“Since we have three equations with a fixed ratio for the variables and constants, this means that there are three dependent, consistent equations, having three equivalent planes and therefore, there are infinite solutions in a plane.
“Geometrically, there are three equivalent planes, each given by the equations. Therefore, there are infinite solutions to the equations in a plane.”
“There is a fixed ratio between the variables and constants to equation () and equation () only.”
(Third equation has same constant, different variables)
“Since we have two equations with a fixed ratio for the variables and constants and one equation with no fixed ratio but with the same constant to another equation, this means that there are three dependent, consistent equations, having two equivalent planes with one oblique plane, and therefore, there are infinite solutions in a line.
“Geometrically, there are two equivalent planes, with one given by the equations () and () and the third plane (given by equation (_), cuts through the equivalent planes obliquely. Therefore, there are infinite solutions to the equations in a line.”
“There is a fixed ratio between the variables and constants to equation () and equation () only.”
(Third equation has same variables, different constant)
“Since we have two equations with a fixed ratio for the variables and constants and one equation with no fixed ratio but with the same variables to another equation, this means that there are three dependent, inconsistent equations, having two equivalent planes with one parallel plane, and therefore, there are no solutions.
“Geometrically, there are two equivalent planes, with one given by the equations () and () and one parallel plane given by equation (_). Therefore, there are no solutions to the equations.”
“There is a fixed ratio between the variables only and not the constants for all 3 equations.”
“Since we have three equations with a fixed ratio for the variables, this means that there are three dependent, inconsistent equations, and therefore, there are no solutions.”
“Geometrically, there are three parallel planes (given by the equations). Therefore, there are no solutions to the equations.”
“There is a fixed ratio between the variables only and not the constants for equation () and equation () only.”
“Since we have two equations with a fixed ratio for the variables and one equation with no fixed ratio, this means that there are two dependent, inconsistent equations, having two parallel planes with one oblique plane, and therefore, there are no solutions.”
“Geometrically, there are two parallel planes (given by equation () and ()) and the third plane (given by equation (_), cuts through both planes obliquely. Therefore, there are no solutions to the equations.”
Internal structure (Main 3 cases)
- Write equations from question, labelling each one and defining the x, y and z variables.
- There is no fixed ratio between the variables and constants for the 3 equations. Therefore, there are 3 possible cases for the equations.
- Independent equations
- Full dependence between variables and constants
- Partial dependence between variables and constants
- I will check for a unique solution on my graphics calculator
(If equations solve, they are CASE 1)
(If equations do not solve, attempt to solve them)
(If equations are equal to each other, they are CASE 2)
(If equations contradict each other, they are CASE 3)
CASE 1
- It is possible to find a unique solution, hence, the equations are independent and consistent.
- Write the unique solutions found in context to the question (eg. number of units sold)
- Geometrically, these three equations represent three planes intersecting at a point, giving a unique solution
CASE 2
- If results of equations are equal to each other when attempting to solve
- It is not possible to find a unique solution, hence, I will check for dependency between equations
- Show equations are the same (equation 4 = equation 5) or (equation 4 = a multiple of equation 5)
“Since these two equations are (identical, or a multiple of each other), then there can be no unique solution for x and y (and hence, z), only a linear relationship between the variables.”
- Show equations are equal to each other 0 = 0
“This is a true statement, indicating the equations () and () are dependent, giving infinite solutions. When attempting to solve algebraically and a result of 0 = 0 occurs, it’s not possible to find a unique solution.”
“Geometrically, each equation represents a plane, and the three planes intersect along a common line (the hinge) where the solutions lie.”
- Simplify to show the relationship between the equations
“Neither equation () and () is a multiple of another, but it may be noted that (relationship). Hence, equation () (add/subtract) equation () gives equation (_), for the variables and constants. Therefore, the equations are dependent, and any two equations can be written in terms of the other. (Essentially, we have 2 independent equations with 3 variables, yielding a linear relationship between the variables.)”
context - It is possible to use up all the variables in many different ways
CASE 3
- If results of equations are not equal to each other when attempting to solve
- It is not possible to find a unique solution, hence, I will check for dependency between equations
- Find equations 4 & 5
“There is a fixed ratio between equation () and () for the variables only and not the constants. Hence, there can be no solution for the set of 3 equations. The equations are inconsistent with each other.”
“I will check for the dependency between the variables in the equations”
- Show equations are the same (equation 4 = equation 5) for the variables only and not the constants
“Since these two equations are identical for the variables only and not the constants, there can be no solution for x and y (and hence, z).”
- Simplify to show the relationship between the equations
“Hence, there is a dependency between the 3 equations for the variables only, such that () equation () (add/subtract) () equation () yields equation (_). Thus, there are no solutions for the set of 3 equations.”
- Show equations are not equal to each other 0 ≠ 2
“This is a false statement, indicating the equations () and () are inconsistent, giving no solutions. When attempting to solve algebraically and a result of 0 ≠ 2 occurs, it’s not possible to find a unique solution.”
“Geometrically, these equations represent three planes, such that a pair of planes intersect in a hinge line and the other pairs intersect in two other hinge lines, all three of which are parallel. Each plane is parallel to the intersection of the other two planes.”
“Graphically…”
In context - It is not possible to use up all the variables
In context - It is not possible to use up all the variables
“In context, with the new proportions, it is not possible for the (person) to meet the requirements for each (variable) exactly. To find a solution they could either try to get a different coefficent of (variable) or change the (total of an equation). However, getting more (variable) (one or more) will only change the constants in each equation and there still may be no solution. Hence, this will not solve the problem. The alternative is to change one of the (total of an equation). The (total of an equation) for (equation) can be slightly (increased/decreased) by (), as to find a solution. For example, (person) could sell (numbers of variables 1, 2 and 3) which would make a total of () OR (numbers of variables 1, 2 and 3) which would make a total of (_)”
In context - It is possible to use up all the variables in many different ways
“The equations are dependent, and therefore any two variables can be written in terms of the other.”
“For example, in terms of parameter (x,y,z), (write coordinates of each point in terms of x, y or z). Or, using parameter t, where t is a whole number, we have (write coordinates of each point in terms of t).”
“In context, with the new proportions, it is possible for the (person) to meet the requirements for each (variable) in many ways. They must however have a non-negative integer (whole number) number of (variable). So, with restrictions, there are many ways the (person) could sell their (variable) to meet all the requirements in the new situation. In particular, any values for x, y and z, where x, y and z are positive integers, and x = (equation 1) and y = (equation 2) and (parameters for each x, y, z value).”
“For example, (person) could sell (numbers of variables 1, 2 and 3) which would make a total of () OR (numbers of variables 1, 2 and 3) which would make a total of ().”
“Using the TABLE function in Graphics Calculator and typing in the general solution setting Y1 > Equation for x
Y2 > Equation for y
Y3 > Equation for z.
For every whole number between the value of z between 0 and (end value, taken from domain max constraint for the chosen parameter), inclusive, the combination that generates the (total value desired, eg. weekly sales of $2200) can be found. The number of (x variable) will vary between () and () (inclusive) and will be multiples of (). The number of (y variable) will vary between () and () (inclusive) and will be multiples of (). And the number of (z variable) will vary between () and () (inclusive) and will be multiples of (_).”
