For what natural number of x and y the equation 2x+5y-7=0 has unique solutions? If the values are taken from collection of real numbers do you think the equation has unique solution
clearly, x=1 and y=1 is a solution.
It is clearly the only solution, since using any other natural numbers (which are positive integers) produces 2x+5y which is greater than 7.
If you include all real numbers, the graph is a line, and any point on the line is a solution. It happens that the only point with positive integer coefficients is (1,1).
To find the natural numbers x and y for which the equation 2x + 5y - 7 = 0 has unique solutions, we need to determine the conditions under which the equation is a straight line with a unique intersection point.
When we rewrite the equation in slope-intercept form, y = mx + c, we get:
5y = -2x + 7
y = (-2/5)x + 7/5
From this form, we can see that the coefficient of x (-2/5) represents the slope of the line. If the slope is not equal to zero, the line is not parallel to the y-axis, and thus, it will have a unique intersection point with the y-axis.
Therefore, the equation 2x + 5y - 7 = 0 will have a unique solution for any natural numbers x and y.
Now, if we consider the values of x and y taken from the collection of real numbers, we need to examine the slope of the line to determine if it has a unique intersection point with the y-axis.
In this case, the equation 2x + 5y - 7 = 0 still has a non-zero slope (-2/5). Hence, the line will intersect the y-axis at a unique point for any real values of x and y.
Thus, whether the values of x and y are taken from the collection of natural numbers or real numbers, the equation 2x + 5y - 7 = 0 will always have a unique solution.