Straight lines are lines on the Cartesian Plane that extend forever in both directions. If we ignore for a moment the special cases of horizontal and vertical lines, straight lines will cross both the $x$x-axis and the $y$y-axis, possibly at the same point (called the origin).
Here are some examples:
The word intercept in mathematics refers to a point where a line, curve or function crosses or intersects with the axes.
Consider what happens as you move up or down along the $y$y-axis. You eventually reach the origin ($\left(0,0\right)$(0,0)) where $y=0$y=0. Now, if you move along the $x$x-axis in either direction, the $y$y value is still $0$0.
Similarly, consider what happens as you move along the $x$x-axis. You eventually reach the origin where $x=0$x=0. Now, if you move along the $y$y-axis in either direction, the $x$x value is still $0$0.
So, two important properties are:
We can use these properties to calculate or identify $x$x and $y$y intercepts for any line, curve or function.
The $x$x intercept occurs at the point where $y=0$y=0.
The $y$y intercept occurs at the point where $x=0$x=0.
Find the $x$x and $y$y intercepts for the following lines.
$y=3x$y=3x
Think: The $x$x intercept occurs when $y=0$y=0. The $y$y intercept occurs when $x=0$x=0.
Do: When $x=0$x=0, $y=3\times0$y=3×0 = $0$0
This means that this line passes through $\left(0,0\right)$(0,0), the origin. The $x$x and $y$y intercept occur at the same point!
This particular form of a straight line $y=mx$y=mx always passes through the origin. (Test it out on the applet below)
$y=4x-7$y=4x−7
Think: The $x$x intercept occurs when $y=0$y=0. The $y$y intercept occurs when $x=0$x=0.
Do: When $x=0$x=0
$y=4\times0-7$y=4×0−7 = $-7$−7 So the $y$y intercept is $-7$−7
When $y=0$y=0
$0$0 | $=$= | $4x-7$4x−7 |
$7$7 | $=$= | $4x$4x |
$\frac{7}{4}$74 | $=$= | $x$x |
So the $x$x intercept is $\frac{7}{4}$74
This form of a straight line $y=mx+b$y=mx+b, always has $y$y intercept of $b$b.
The $x$x intercept is easy to work out after that (substitute $y=0$y=0).
$2y-5x-10=0$2y−5x−10=0
Think: The $x$x intercept occurs when $y=0$y=0. The $y$y intercept occurs when $x=0$x=0.
Do: When $x=0$x=0, the $5x$5x term disappears. This leaves us with:
$2y-10$2y−10 | $=$= | $0$0 |
$2y$2y | $=$= | $10$10 |
$y$y | $=$= | $5$5 |
So the $y$y intercept is $5$5
When $y=0$y=0, the $2y$2y term disappears. This leaves us with:
$-5x-10$−5x−10 | $=$= | $0$0 |
$-5x$−5x | $=$= | $10$10 |
$x$x | $=$= | $-2$−2 |
So the $x$x intercept is $-2$−2
Let's have a look at some worked solutions.
What is the $x$x intercept of the line $-3x+4y=-27$−3x+4y=−27?
What is the $y$y intercept of the line with equation $-5x+3y=27$−5x+3y=27?
A line has equation $-6x+2y-12=0$−6x+2y−12=0.
a) Find the y value at the point where $x=0$x=0.
b) Find the $x$x value at the point where $y=0$y=0.
c) Use these two points to graph the line.