Linear equations; how to write an equation from given coordinates?

Question

A straight line goes through the points $( 0, 1 )$, $( 2, 7 )$ and $( 4, 13 )$ and I need to write the equation of this straight line. How do you write equations?

I know you have it's usually $y = x \text{(something)} + \text{something}$.

Is it trial and error or is there a better way? We just got introduced to linear equations so it's new, I know how to work out the coordinates to make a line but I'm not sure on how to write the equation from coordinates.

Answer 1

1. Find the gradient $m$ of the line by using two of the points given. Recall that gradient can be calculated by: $\dfrac{y_2-y_1}{x_2-x_1}$
2. With a gradient and one point, you can use $y=mx+c$ by substituting the point into $x$ and $y$ to find the value of $c$, the y-intercept.
3. Lastly, take $y=mx+c$ and substitute $m$ and $c$ by the values you obtained in steps 1 and 2.

Check the points to see if all of them satisfy the equation of the line.

Answer 2

First, you only need two of those points. Let's take the first two, and make the identifications: $$(x_1,y_1) = (0,1)$$ $$ (x_2,y_2) = (2,7)$$

Then we use the slope formula: $$m = \dfrac{y_2-y_1}{x_2-x_1}$$

Finally, we plug all this into the "point slope" formula, and simplify as necessary: $$y-y_1 = m(x-x_1)$$