Question

Write the equation for the following relation. Q= {(x,y): (4,5), (8,7), (12,9), (16,11),...}​

Answer 1

There's really no way to solve this problem apriori (unless you can use tools like lagrange polynomials to interpolate points), so I'll just tell you how to approach problems like this.

First of all, we may try to see if the dependence is linear: the input is always increased by 4 (4, 8, 12, 16,...) and the output increases by 2: (5, 7, 9, 11). So, the answer is yes.

Now that we know that these points lay on a line, we can conclude the exercise in several ways:

1. We already know that the slope is 1/2 (4 units up in the x direction correspond to 2 units up in the y direction). So, we only need the y-intercept. If we go back one step, we can see that the next point would be (0, 3) (I decreased the x coordinate by 4 and the y coordinate by 2). So, the y intercept is 3, and the equation of the line is $y=\frac{x}{2}+3$
2. We can use the equation of the line passing through two points:

$\dfrac{x-x_2}{x_1-x_2}=\dfrac{y-y_2}{y_1-y_2}$

Plug in two points of your choice and you'll get the same answer.

Of course, as a third alternative, you could just have eyeballed the answer: the fact that x grows twice as fast as y could have hinted the x/2 part, and then you could have seen that y is always 3 more than half of x, again leading to y=x/2+3.