Beginning Activity Beginning Activity 2: Linear Equations in Two Variables
1.
Find integers \(x\) and \(y\) so that \(2x + 6y = 25\) or explain why it is not possible to find such a pair of integers.
2.
Find integers \(x\) and \(y\) so that \(6x - 9y = 100\) or explain why it is not possible to find such a pair of integers.
3.
Notice that \(x = 2\) and \(y = 1\) is a solution of the equation \(3x + 5y = 11\text{,}\) and that \(x = 7\) and \(y = -2\) is also a solution of the equation \(3x + 5y = 11\text{.}\)
(a)
Find two pairs of integers \(x\) and \(y\) so that \(x > 7\) and \(3x + 5y = 11\text{.}\) (Try to keep the integer values of \(x\) as small as possible.)
(b)
Find two pairs of integers \(x\) and \(y\) so that \(x \lt 2\) and \(3x + 5y = 11\text{.}\) (Try to keep the integer values of \(x\) as close to 2 as possible.)
(c)
Determine formulas (one for \(x\) and one for \(y\)) that will generate pairs of integers \(x\) and \(y\) so that \(3x + 5y = 11\text{.}\)
The two formulas can be written in the form \(x = 2 + km\) and \(y = 1 + kn\text{,}\) where \(k\) is an arbitrary integer and \(m\) and \(n\) are specific integers.
4.
Notice that \(x = 4\) and \(y = 0\) is a solution of the equation \(4x + 6y = 16\text{,}\) and that \(x = 7\) and \(y = -2\) is a solution of the equation \(4x + 6y = 16\text{.}\)
(a)
Find two pairs of integers \(x\) and \(y\) so that \(x > 7\) and \(4x + 6y = 16\text{.}\) (Try to keep the integer values of \(x\) as small as possible.)
(b)
Find two pairs of integers \(x\) and \(y\) so that \(x \lt 4\) and \(4x + 6y = 16\text{.}\) (Try to keep the integer values of \(x\) as close to 4 as possible.)
(c)
Determine formulas (one for \(x\) and one for \(y\)) that will generate pairs of integers \(x\) and \(y\) so that \(4x + 6y = 16\text{.}\)
The two formulas can be written in the form \(x = 4 + km\) and \(y = 0 + kn\text{,}\) where \(k\) is an arbitrary integer and \(m\) and \(n\) are specific integers.