The discussion revolves around the nature of solutions for a specific linear equation involving two variables, ##a## and ##b##. Participants explore the implications of having one equation with two variables, questioning the common understanding that such systems are underdetermined and typically yield infinitely many solutions. The conversation delves into the uniqueness of solutions based on different interpretations and conditions applied to the variables.
Participants express differing views on the nature of solutions for the equation, with no consensus reached on whether it leads to a unique solution or infinitely many solutions. The discussion remains unresolved regarding the implications of variable conditions on the solutions.
Participants reference various conditions (such as integer restrictions) that may affect the nature of the solutions, but these conditions are not universally accepted or agreed upon in the discussion.
I may be missing something here, but what "unique solution" are you referring to? This is a linear expression so it should have an infinite amount of solutions, as there are no discontinuites in its graph.Mr Davis 97 said:I have the linear equation ##a + (2a - b) \sqrt{2} = 4 + \sqrt{2}##. I commonly hear that for linear equations, if we have two variables and one equation, then the system is undertermined and there are infinitely many solutions. However, how does that jive with this example, where there is one unique solution after comparing coefficients?
It seems to me that you are restricting your solution to the integers. Solving your equation with respect to a gives a=\frac{(b+1)\sqrt{2} + 4}{1+2\sqrt{2}}, which gives one value of a for each value of b you insert in the formula.Mr Davis 97 said:I have the linear equation ##a + (2a - b) \sqrt{2} = 4 + \sqrt{2}##. I commonly hear that for linear equations, if we have two variables and one equation, then the system is undertermined and there are infinitely many solutions. However, how does that jive with this example, where there is one unique solution after comparing coefficients?
It depends on what type of equation you're working with.Mr Davis 97 said:I have the linear equation ##a + (2a - b) \sqrt{2} = 4 + \sqrt{2}##. I commonly hear that for linear equations, if we have two variables and one equation, then the system is undertermined and there are infinitely many solutions. However, how does that jive with this example, where there is one unique solution after comparing coefficients?
Mr Davis 97 said:if we have two variables and one equation, then the system is undertermined and there are infinitely many solutions.