1: Lines
The algebra and geometry of intersecting lines
This section is also part of a related site on AS91587
Finding the intersection of two lines in two dimensions is a good introduction to the main part of this topic. The key ideas for intersecting lines extend to intersecting regions for inequalities.
There are several ways to represent the equation of a line. The one you are most familiar with is probably y = mx + c .
All of the pairs of points (x,y) which make the equation true form the line.
In this form, the gradient of the line is m, and the y-axis intercept is c.
A horizontal line with gradient zero is written y = c (we can think of it as having m = 0).
Vertical lines cannot be represented in this form.
Another form to write an equation for a line is Ax + By = C .
This is the general form for the equation of a line, and can represent lines vertical lines.
A technical requirement is that A and B cannot both be zero.
This form (and later an extension to a third variable, z) will be useful when using calculators and computers to solve systems of equations.
Solving for intersecting lines
There are several ways to find the intersection of two lines. All require finding the point(s) at which both line equations are true.
Graphical Method / Tabular Method
Sometimes graphs of the two lines will make it clear where the intersection lies.
You should check to make sure that the point lies on both lines by substituting it into the equations and checking they are both true.
Making a table of values on the lines might also show a point that is common to both lines.
Elimination Method
When the equations are in the form Ax + By = C, sometimes in is easier to form a new equation that is independent of x.
Example
Find the intersection of the lines (4,5)
3x + 5y = 37
2x + 3y = 23
Solution
Form a new equation, by taking 2 times the first equation.
6x + 10y = 74
Form a new equation, by taking 3 times the second equation.
6x + 9y = 69
Notice that these were chosen to get the same coefficient of x in both equations. Subtract one from the other.
(6x + 10y) - (6x + 9y) = 74 - 69
y = 5
Substitute back to find
2x + 3 × 5 = 23
2x = 8
x = 4
Substitution Method
Write both of the equations in the form y = mx + c.
Put them equal to each other, and solve for x.
Use the x-value to find the value of y.
Example
Find the intersection of the lines:
2x +5y = 20
5x + 2y = 29
Solution
Method 1
Rearrange the first equation.
5y = 20 - 2x
y = 4 - 0.4 x
Rearrange the second equation.
2y = 29 - 5x
y = 14.5 - 2.5x
Now we can see (since both expressions involving x equal y)
4 - 0.4x = 14.5 - 2.5x
Rearranging
2.1x = 10.5
x = 10.5 / 2.1 = 5
Substitute this x-value back into either line (or both, to be sure)
y = 4 - 0.4 × 5 = 2
The point of intersection is (x,y) = (5,2).
Method 2
Rearrange the first equation, y = 4 - 0.4x.
Substitute this expression for y into the second equation.
2y + 5x = 29
2(4 - 0.4x) + 5x = 29
Notice that this equation no longer has y in it.
Simplify and solve for x.
8 +4.2x = 29
4.2x = 21
x = 21 / 4.2 = 5
Now substitute back to find y = 2.
Exercises
This should really be straightforward revision of skills built over NCEA levels 1 and 2.
For every pair of lines, find their intersection point.
Find all 15 points.
Line A: y = x + 2
Line B: 2x - y = 5
Line C: x + y = 2
Line D: 3x - y + 1 = 0
Line E: 2y = x + 16
Line F: 5x - 2y = 38