In this explainer, we will learn how to represent a straight line in the two-intercept form using the - and -intercepts.

We should already be familiar with some of the different forms in which the equation of a straight line can be given. Those that will be useful here are recalled below.

Name | Equation | Key Information |
---|---|---|

Slopeβintercept form | = slope of the line = -intercept | |

Pointβslope form | = slope of the line = coordinates of any point on the line | |

Standard form | For constants , , and | |

General form | For constants , , and |

The various forms of the equation of a straight line are useful in different contexts. They may reveal different features of the line and its graph or allow us to find the equation of a straight line given a particular set of information. We will now introduce another form in which the equation of a straight line can be given: the two-intercept form.

### Definition: The Two-Intercept Form of the Equation of a Straight Line

The two-intercept form of the equation of a straight line that intercepts the -axis at and intercepts the -axis at is where and .

The two-intercept form of the equation of a straight line is useful because it enables us to identify the - and -intercepts directly, with no rearrangement required. We could then use the coordinates of these two points for other purposes, such as to plot the line or to calculate its slope. We will now consider the derivation of the two-intercept form of the equation of a straight line.

We recall that, in general, the slope of a line passing through the points with coordinates and is

Now consider a straight line that intercepts the -axis at and the -axis at , as shown in the sketch below.

Substituting the coordinates of these two points into the slope formula, we find that the slope of this line is

Next, we recall the pointβslope form of the equation of a straight line, . Substituting and for gives

Distributing the parentheses and then dividing both sides by gives

Finally, adding to each side of the equation gives

This is the two-intercept form of the equation of the straight line that intercepts the -axis at and the -axis at . We see that and are the denominators of the two fractions; hence, given the equation of a straight line in this form, we can immediately identify its - and -intercepts. Alternatively, given the - and -intercepts of a line, we can write down its equation in this form.

There are three notable exceptions we should be aware of: the equations of horizontal lines and those of vertical lines cannot be expressed in two-intercept form. Each of these types of lines are parallel to one of the coordinate axes and hence do not intercept this axis. We recall that the equation of a horizontal line is and the equation of a vertical line is , for an arbitrary constant . Neither of these equations can be rearranged into two-intercept form. The third exception is the line that passes though the origine for which the constant term is 0, so it cannot be written in the intercept form.

We will now consider a series of examples in which we use this form of the equation of a straight line. In our first example, we will practice writing the equation of a straight line in two-intercept form when given the coordinates of the points at which it intercepts each axis.

### Example 1: Writing the Equation of a Straight Line in Two-Intercept Form given Its π₯- and π¦-Intercepts

If a straight line intercepts the -axis at and intercepts the -axis at , write the equation of the straight line in two-intercept form.

### Answer

We recall first that the two-intercept form of the equation of a straight line that intercepts the -axis at and intercepts the -axis at is

As this line intercepts the -axis at , the value of is 6. As the line intercepts the -axis at , the value of is 5. Substituting and into the equation above gives

We have seen how to write the equation of a straight line in two-intercept form given both its - and -intercepts. We will now consider the reverse of this: how to identify the - and -intercepts of a line given its equation in two-intercept form. In the example, we will now consider, we will need to exercise care as one of the intercepts has a negative value.

### Example 2: Finding the π₯- and π¦-Intercepts When Given the Equation of a Straight Line in Two-Intercept Form

List the coordinates of the -intercept and the -intercept of the line .

### Answer

We note that the equation of this line looks very similar to the two-intercept form of the equation of a straight line: where the coordinates of the -intercept of the line are and the coordinates of the -intercept are . However, in the equation we have been given, there is a subtraction sign, rather than an addition sign, between the two terms on the left-hand side.

We may find it helpful to manipulate the equation slightly so that it more closely aligns with the two-intercept form. The term ββ is equivalent to ββ, so the equation can be rewritten as

Although we would not normally choose to leave a negative value in the denominator of a fraction, this form now directly mirrors the two-intercept form. We can, therefore, determine the values of and by considering the denominators of the two fractions. The value of is 3 and the value of is , so the coordinates of the - and -intercepts of the line are and respectively.

It is not strictly necessary to convert the equation of a line into the two-intercept form in order to determine its - and -intercepts. As the coordinates of the -intercept are , another possible method is to substitute the value 0 for in any form of the equation of a straight line and then solve to determine the value of . In the same way, substituting 0 for would enable us to solve for and determine the -intercept.

As we saw in the previous example, extra care needs to be taken when finding the - or -intercepts from the two-intercept form if either of them has a negative value. Similarly, we must take extra care if either intercept has a fractional value. For example, consider the straight line with equation .

We may mistakenly believe that this line is in two-intercept form with and and hence conclude that the line intercepts the -axis at and the -axis at . However, if we compare this form with the two-intercept form of the equation of the straight line, we see that the - and -terms should be divided by the constants that represent the - and -intercepts, not multiplied by them. Although this is an untidy form that we would not usually choose, we can express as , as we see below:

In the same way, can be expressed as . Thus, the equation can be rewritten as and we now see that the line intercepts the -axis at and the -axis at .

Sometimes, the equations we are given will be in other forms, such as the pointβslope or slopeβintercept forms. Converting between the various forms of the equation of a straight line is an essential skill because the different forms are useful for identifying different properties of the line.

In our next example, we will see how we can rearrange the equation of a straight line given in slopeβintercept form into two-intercept form.

### Example 3: Converting the Equation of a Straight Line into Two-Intercept Form

Write the equation of the line in the two-intercept form.

### Answer

The equation of this line has been given in slopeβintercept form, , where represents the slope of the line and represents the -intercept. We recall that the two-intercept form of the equation of a straight line is where is the -intercept of the line and is the -intercept. We are, therefore, required to rearrange the equation given into the desired form.

We begin by adding to each side of the equation, giving

We have now collected the - and -terms on one side of the equation with the constant term on the other. For the equation to be in two-intercept form, the constant term must be equal to 1. Dividing both sides by 6 achieves this and gives

Finally, we simplify the first fraction by canceling a common factor of 2 in the numerator and denominator:

It is important to ensure we do not confuse the two-intercept form of the equation of a straight line with the standard form . In the previous example, we had rearranged the equation of the line to give at one stage in our working. This is not in two-intercept form as the constant on the right-hand side of the equation is not equal to 1. It is important to remember that if the constant on the right-hand side of the equation is equal to a general constant , we must divide every term in the equation by before we can use it to determine the - and -intercepts of the line.

Although it was not required in the previous example, we could use the two-intercept form of the equation of the line to identify its - and -intercepts. The term reveals that the line intercepts the -axis at , and the term reveals that the line intercepts the -axis at . As the equation of the line was originally given in slopeβintercept form, we could have also identified the -intercept from this form of the equation but the -intercept would not have been as immediately obvious.

In our next example, we will demonstrate how to find the equation of a straight line in the two-intercept form from its graph.

### Example 4: Writing the Two-Intercept Form of the Equation of a Straight Line given Its Graph

Write the equation represented by the graph shown. Give your answer in the form .

### Answer

We should recognize that the form in which our answer has been requested is the two-intercept form of the equation of a straight line: where are the coordinates of the -intercept of the line and are the coordinates of the -intercept.

From the graph, we identify that the line intercepts the -axis at the point and intercepts the -axis at the point . Hence, the values of and are 3 and 9 respectively. Substituting and into the two-intercept form gives

We saw in our earlier derivation of the two-intercept form of the equation of a straight line that the slope of a line given in this form is equal to . We can demonstrate this another way by rearranging the equation of a straight line given in two-intercept form into the slopeβintercept form . Starting with the equation , we isolate by first subtracting from each side, giving

Multiplying both sides by gives

This equation is now in slopeβintercept form and we recall that the coefficient of represents the slope of the line. Therefore, we see again that the slope of the line is . This result is useful because it allows us to identify the slope of a line given in two-intercept form without performing any rearrangement of its equation.

### Theorem: The Slope of a Line Given in Two-Intercept Form

The slope of a straight line given in two-intercept form , where are the coordinates of the -intercept of the line and are the coordinates of the -intercept, is .

Let us consider one final example in which we calculate both intercepts and the slope of a line given in two-intercept form.

### Example 5: Identifying the π₯- and π¦-Intercepts and the Slope of a Straight Line Given in Two-Intercept Form

The graph of the equation is a straight line.

- What are the coordinates of the -intercept of the line?
- What are the coordinates of the -intercept of the line?
- What is the slope of the line?

### Answer

We recognize that the equation of this straight line has been given in two-intercept form: where are the coordinates of the -intercept of the line and are the coordinates of the -intercept. By comparing the equation of our line with the general equation above, we see that and . Hence, the coordinates of the -intercept are and the coordinates of the -intercept are .

We now recall that the slope of a line given in two-intercept form is . Using the values we have already determined for and , the slope is equal to

Hence, the coordinates of the -intercept are , the coordinates of the -intercept are , and the slope is .

Let us finish by recapping some key points from this explainer.

### Key Points

- The two-intercept form of the equation of a straight line that intercepts the -axis at and intercepts the -axis at is where and .
- The slope of a straight line given in two-intercept form is .
- The equation of a straight line given in another form can be converted into two-intercept form by rearranging, and vice versa.