The first step in finding of a linear pattern is to see how much the output (y) increases every time, in this case we see that it goes up by 4 every time, regardless of what number we put into this input/output machine, it will always be 4 less or more the ones “around” it. Anyway, this increase of 4 signifies the coefficient that our rule will use, what we multiply our variable by.

The next step is to put your input number in the rule you have so far and see what else you have to add to it to get to your output number, in this case 4 x 1 would still be 4 and what you have to add to it to get to 7 would be 3, this works with any pair here, for our 3rd pair, 4 x 3 = 12 and again we have to add 3 to get our output number of 15. Because this is consistent across this whole pattern, our rule is definitely 4x + 3 = y.

To visualize this pattern further you can graph them and see where each point is in relation to each other where the input is your x coordinate and the output is your y coordinate.

When it’s all graphed out you can see that everything is in a straight line, this is because when you always go up by the same number each time nothing really ever changes in the direction you go, all linear patterns (rules with degree of 1) work like this and are kind of why they are called linear, because they make a straight line when you graph it.

When graphing these points manually, connecting the dots with lines or not comes down to what kind of variables you are working with, continuous or discrete, the difference between the two is one can be split/have decimals and one can’t, think of people as an example, you can’t necessarily split a person to have 1.86 people. In those cases dots don’t get connected like they do here, but when working with any kind of unit as a variable, (this means numbers too) you can connect the dots together with a line because the space the line takes where decimals would be are possible because it’s just a regular number that can be split.