Table of Contents
- 1 How do you prove that an equation has infinite solutions?
- 2 Does every Pell equation have a solution?
- 3 How do you find the fundamental solution of the Pell equation?
- 4 Which of the following systems of equations has infinitely many solutions?
- 5 What does it mean if an equation has an infinite number?
- 6 Which pair of linear equations will always have unique solutions?
How do you prove that an equation has infinite solutions?
We can identify which case it is by looking at our results. If we end up with the same term on both sides of the equal sign, such as 4 = 4 or 4x = 4x, then we have infinite solutions. If we end up with different numbers on either side of the equal sign, as in 4 = 5, then we have no solutions.
Does every Pell equation have a solution?
(4) Pell’s equation always has nontrivial solutions. The fundamental solution is x y = { [ a 0 , a 1 , … , a k − 1 ] if k is even [ a 0 , a 1 , … , a 2 k − 1 ] if k is odd .
What does it mean when an equation has infinitely many solutions?
It is impossible for the equation to be true no matter what value we assign to the variable. Infinite solutions would mean that any value for the variable would make the equation true.
How do you find the generalized Pell equation?
x1 − y1 √ d = (x2 − y2 √ d)(x − y √ d). We saw in the previous section that Pell’s equation has a nontrivial solution. Using a nontrivial solution of Pell’s equation we will describe a method to write down all the solutions of a generalized Pell equation x2 − dy2 = n, where n is a nonzero integer.
How do you find the fundamental solution of the Pell equation?
solving Pell’s equation and minimizing x satisfies x1 = hi and y1 = ki for some i. This pair is called the fundamental solution. Thus, the fundamental solution may be found by performing the continued fraction expansion and testing each successive convergent until a solution to Pell’s equation is found.
Which of the following systems of equations has infinitely many solutions?
dependent system
A dependent system has infinitely many solutions. The lines are exactly the same, so every coordinate pair on the line is a solution to both equations.
How do you prove an equation has an infinite solution?
An equation will produce an infinite solution if it satisfies some conditions for infinite solutions. An infinite solution can be produced if the lines are coincident and they must have the same y-intercept. The two lines having the same y-intercept and the slope, are actually the exact same line.
What are infinite solutions?
What are Infinite Solutions? The number of solutions of an equation depends on the total number of variables contained in it. Thus, the system of the equation has two or more equations containing two or more variables. It can be any combination such as
What does it mean if an equation has an infinite number?
It means that if the system of equations has an infinite number of solution, then the system is said to be consistent. As an example, consider the following two lines. These two lines are exactly the same line. If you multiply line 1 by 5, you get the line 2.
Which pair of linear equations will always have unique solutions?
A consistent pair of linear equations will always have unique or infinite solutions. Example 1) Here are two equations in two variables. a1x + b1y = c1 ——- (1) a2x + b2y = c2 ——- (2)