Do two random lines, y=ax + b and y=cx + d, intersect? How do we work this out and if they do, where do they intersect?

Let us use Line 1 and Line 2 as an example.

Line 1:  y = 2x + 3

Line 2: y = -0.5x + 7

Step 1 - Try and plot the two functions on a x, y coordinate axes.

Remember y = mx + c. Using Line1, we are given the y intercept (c) which is 3. We are also given the gradient (m) which is 2.

We can make one mark at the point (0,3) which corresponds to the y intercept. Another way to think of this: what is y when x=0?

Following on from this: what is x when y = 0 ? To work this out just substitute y = 0 into Line 1. After rearranging (to make it x = all other stuff )  we find that x = -3/2. We now have another point to mark on our axes (-3/2, 0).

Now draw a line between both marks and continue the line in both directions! Repeat for Line 2. It is not vital to plot the lines, but by doing it correctly we can already answer by inspection of the graph " Do these lines intersect?". By doing the graph axis perfectly with sufficient divisions you can even read of the point of intersection (not recommended). In this case the lines do intersect. So what’s next? Let’s prove it using algebra.

Step 2 - What are the coordinates of the point where Line 1 and Line 2 intersect?

Let’s call the point of intersection P with the coordinates (x,y) on our graph.

A good way to think about this is to realise that Line 1 and Line 2 both contain the two same variables, x and y.

At the point P, it is the only position where those two variables x & y must satisfy both Lines!!

The way to prove this is by making the two Line equations equal to each other.

They both start with y = (rest of stuff). At point P, y must be the same for both so we can literally make them equal to each other.

y=y so therefore Line 1 = Line 2

2x + 3 = -0.5x + 7    ( We now have one equation and one variable! We can solve this)

Rearrange (So x = all other stuff) and we find that x = 1.6!  We have the first coordinate of P (1.6, y)

Now let’s find y! Pick either of the Lines, it does not matter because at point P, x is going to be the same in both.

Picked Line 1: y = 2x + 3

Substitute in our lovely x value we have just worked out (x = 1.6). And we get:

y = 2*(1.6) + 3

y = 6.2

Now we have the second coordinate of P!

P = point of intersection of Lines 1 & 2 = (1.6, 6.2)