Linear Equations

NZ Level 7 (NZC) Level 2 (NCEA)

Intersections of Lines

Lesson

Because lines extend forever in both directions, they will intersect somewhere (unless they are parallel).

Now, when 3 or more lines all pass through the same point we give those lines a special name: they are called **concurrent **lines.

The point of intersection is called the "point of concurrency", labelled point **P** below.

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Where two lines intersect, they share a common point. At this point both the $x$`x` and $y$`y` values of the equations are equal.

If one line has equation $y=2x+3$`y`=2`x`+3 and another has equation $y=x+6$`y`=`x`+6 then the point of intersection is where both the $y$`y`'s are the same value. If they are the same value, then we can say that:

$2x+3$2`x`+3 is equal in value to $x+6$`x`+6 (at the point where they meet)

Then:

$2x+3=x+6$2`x`+3=`x`+6

Solving this equation for $x$`x`:

$2x-x=6-3$2`x`−`x`=6−3

$x=3$`x`=3

Now that we have $x$`x`, we can find the $y$`y` value at the point of intersection by substituting back into the equation. Which equation should we use? Well since the point of intersection satisfies both equations, we can use either one.

$y=x+6$`y`=`x`+6

$y=3+6$`y`=3+6

$y=9$`y`=9

So these lines cross at the point $\left(3,9\right)$(3,9).

**Testing the solution**:

We used $y=x+6$`y`=`x`+6 to find the $y$`y` value of the point of intersection. Now to test the point, we can substitute $x=3$`x`=3 and $y=9$`y`=9 into the other equation, $y=2x+3$`y`=2`x`+3, and make sure it satisfies both equations.

Substituting $x=3$`x`=3, we get:

$y=2\times3+3$`y`=2×3+3

= $9$9 (same $y$`y` value as we got before)

Using this interactive you can set two different equations of lines and watch how the working out changes in trying to find the point of intersection.

Let's have a look at a worked example.

Consider the graph of the two lines $y=-4x+4$`y`=−4`x`+4 and $y=x-6$`y`=`x`−6.

Loading Graph...

Find the value of $x$

`x`where the lines intersect.Find the value of $y$

`y`where the lines intersect.Hence, state the coordinates of the point of intersection. State your answer in the form $\left(a,b\right)$(

`a`,`b`).

Apply co-ordinate geometry techniques to points and lines

Apply co-ordinate geometry methods in solving problems