# Intercepts

Lesson

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 or maybe through the point where the $x$x and $y$y axes cross (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.

• We can have $x$x intercepts: where the line, curve or function crosses the $x$x axis.
• We can have $y$y intercepts: where the line, curve or function crosses the $y$y axis.

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:

• any point on the $x$x-axis will have $y$y value of $0$0
• any point on the $y$y-axis will have $x$x value of $0$0

We can use these properties to calculate or identify $x$x and $y$y intercepts for any line, curve or function.

Intercepts

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.

### Written Examples

Find the $x$x and $y$y intercepts for the following lines.

##### Question 1

$y=3x$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.

##### Question 2

$y=4x-7$y=4x7

Think:  The $x$x intercept occurs when $y=0$y=0.  The $y$y intercept occurs when $x=0$x=0.

DoWhen $x=0$x=0

$y=4\times0-7$y=4×07 = $-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).

##### Question 3

$2y-5x-10=0$2y5x10=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.

##### Question 4

What is the $x$x-value of the $x$x-intercept of the line $-3x+4y=-27$3x+4y=27?

##### Question 5

What is the $y$y-value of the $y$y-intercept of the line with equation: $-5x+3y=27$5x+3y=27

##### Question 6

Consider the points in the table:

 Time in minutes ($x$x) Temperature in °C ($y$y) 1 2 3 4 5 $6$6 $9$9 $12$12 $15$15 $18$18
1. By how much is the temperature increasing each minute?

2. What would the temperature have been at time 0?

3. Find the algebraic relationship between $x$x and $y$y.

### Outcomes

#### 9D.AG2.01

Determine, through investigation, various formulas for the slope of a line segment or to determine the slope of a line segment or a line