Answer:
ONE SOLUTION
Step-by-step explanation:
When two points on a line are given, the equation of the line is given by the formula:
[tex]$ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} $[/tex]
where [tex]$ (x_1, y_1) $[/tex] and [tex]$ (x_2, y_2) $[/tex] are the points on the line.
Here, the first set of points are: [tex]$ (-1, 3) $[/tex] and [tex]$ (0, 1) $[/tex].
Therefore, [tex]$ (x_1, y_1) = (-1, 3) $[/tex] and [tex]$ (x_2,y_2) = (0, 1) $[/tex].
The line passing through this is given by:
[tex]$ \frac{y - 3}{1 - 3} = \frac{x + 1}{1} $[/tex]
[tex]$ \implies y - 3 = -2x - 2 $[/tex]
∴ 2x + y - 1 =0
Now, for the second line, the points are:
[tex]$ (x_1, y_1) = (1, 4) $[/tex] and [tex]$ (x_2, y_2) = (0, 2) $[/tex].
Therefore, [tex]$ \frac{y - 4}{-2} = \frac{x - 1}{-1} $[/tex]
[tex]$ \implies -y + 4 = -2x + 2 $[/tex]
∴ 2x - y + 2 = 0
Now, to determine the number of solutions the two equations have, we solve these two equations,
Adding Eqn(1) and Eqn(2) we get:
4x = -1
[tex]$ \implies x = \frac{-1}{4} $[/tex]
And [tex]$ y = \frac{3}{2} $[/tex].
Since, we arrive at unique values of 'x' and 'y', we say the lines have only one unique solution.
