Lesson Explainer: Equation of a Straight Line: Two-Intercept Form Mathematics

In this explainer, we will learn how to represent a straight line in the two-intercept form using the π‘₯- and 𝑦-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.

NameEquationKey Information
Slope–intercept form𝑦=π‘šπ‘₯+π‘π‘š = slope of the line
𝑏 = 𝑦-intercept
Point–slope formπ‘¦βˆ’π‘¦=π‘š(π‘₯βˆ’π‘₯)οŠ§οŠ§π‘š = slope of the line
(π‘₯,𝑦) = coordinates of any point on the line
Standard formπ‘Žπ‘₯+𝑏𝑦=𝑐For constants π‘Ž, 𝑏, and 𝑐
General formπ‘Žπ‘₯+𝑏𝑦+𝑐=0For constants π‘Ž, 𝑏, and 𝑐

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 π‘₯-axis at (π‘Ž,0) and intercepts the 𝑦-axis at (0,𝑏) is π‘₯π‘Ž+𝑦𝑏=1.

The two-intercept form of the equation of a straight line is useful because it enables us to identify the π‘₯- and 𝑦-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 (π‘₯,𝑦) and (π‘₯,𝑦) is π‘š=π‘¦βˆ’π‘¦π‘₯βˆ’π‘₯.

Now consider a straight line that intercepts the π‘₯-axis at (π‘Ž,0) and the 𝑦-axis at (0,𝑏), 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 π‘š=0βˆ’π‘π‘Žβˆ’0=βˆ’π‘π‘Ž.

Next, we recall the point–slope form of the equation of a straight line, π‘¦βˆ’π‘¦=π‘š(π‘₯βˆ’π‘₯). Substituting π‘š=βˆ’π‘π‘Ž and (π‘Ž,0) for (π‘₯,𝑦) gives π‘¦βˆ’0=βˆ’π‘π‘Ž(π‘₯βˆ’π‘Ž).

Distributing the parentheses and then dividing both sides by 𝑏 gives 𝑦=βˆ’π‘π‘₯π‘Ž+𝑏𝑦𝑏=βˆ’π‘₯π‘Ž+1.

Finally, adding π‘₯π‘Ž to each side of the equation gives π‘₯π‘Ž+𝑦𝑏=1.

This is the two-intercept form of the equation of the straight line that intercepts the π‘₯-axis at (π‘Ž,0) and the 𝑦-axis at (0,𝑏). We see that π‘Ž and 𝑏 are the denominators of the two fractions; hence, given the equation of a straight line in this form, we can immediately identify its π‘₯- and 𝑦-intercepts. Alternatively, given the π‘₯- and 𝑦-intercepts of a line, we can write down its equation in this form.

There are two 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 𝑦=𝑐 and the equation of a vertical line is π‘₯=𝑐, for an arbitrary constant 𝑐. Neither of these equations can be rearranged into two-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 𝑦-Intercepts

If a straight line intercepts the π‘₯-axis at (6,0) and intercepts the 𝑦-axis at (0,5), 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 π‘₯-axis at (π‘Ž,0) and intercepts the 𝑦-axis at (0,𝑏) is π‘₯π‘Ž+𝑦𝑏=1.

As this line intercepts the π‘₯-axis at (6,0), the value of π‘Ž is 6. As the line intercepts the 𝑦-axis at (0,5), the value of 𝑏 is 5. Substituting π‘Ž=6 and 𝑏=5 into the equation above gives π‘₯6+𝑦5=1.

We have seen how to write the equation of a straight line in two-intercept form given both its π‘₯- and 𝑦-intercepts. We will now consider the reverse of this: how to identify the π‘₯- and 𝑦-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 𝑦-Intercepts When Given the Equation of a Straight Line in Two-Intercept Form

List the coordinates of the π‘₯-intercept and the 𝑦-intercept of the line π‘₯3βˆ’π‘¦2=1.

Answer

We note that the equation of this line looks very similar to the two-intercept form of the equation of a straight line: π‘₯π‘Ž+𝑦𝑏=1, where the coordinates of the π‘₯-intercept of the line are (π‘Ž,0) and the coordinates of the 𝑦-intercept are (0,𝑏). 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 β€œβˆ’π‘¦2” is equivalent to β€œπ‘¦βˆ’2”, so the equation can be rewritten as π‘₯3+π‘¦βˆ’2=1.

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 π‘Ž and 𝑏 by considering the denominators of the two fractions. The value of π‘Ž is 3 and the value of 𝑏 is βˆ’2, so the coordinates of the π‘₯- and 𝑦-intercepts of the line are (3,0) and (0,βˆ’2) respectively.

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

As we saw in the previous example, extra care needs to be taken when finding the π‘₯- or 𝑦-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 5π‘₯+7𝑦=1.

We may mistakenly believe that this line is in two-intercept form with π‘Ž=5 and 𝑏=7 and hence conclude that the line intercepts the π‘₯-axis at (5,0) and the 𝑦-axis at (0,7). However, if we compare this form with the two-intercept form of the equation of the straight line, we see that the π‘₯- and 𝑦-terms should be divided by the constants that represent the π‘₯- and 𝑦-intercepts, not multiplied by them. Although this is an untidy form that we would not usually choose, we can express 5π‘₯ as π‘₯, as we see below: π‘₯=π‘₯Γ·15=π‘₯Γ—51=5π‘₯.

In the same way, 7𝑦 can be expressed as π‘¦οŠ§οŠ­. Thus, the equation 5π‘₯+7𝑦=1 can be rewritten as π‘₯+𝑦=1, and we now see that the line intercepts the π‘₯-axis at ο€Ό15,0 and the 𝑦-axis at ο€Ό0,17.

Sometimes, the equations we are given will be in other forms, such as the point–slope or slope–intercept 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 slope–intercept form into two-intercept form.

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

Write the equation of the line 𝑦=βˆ’2π‘₯+6 in the two-intercept form.

Answer

The equation of this line has been given in slope–intercept form, 𝑦=π‘šπ‘₯+𝑏, where π‘š represents the slope of the line and 𝑏 represents the 𝑦-intercept. We recall that the two-intercept form of the equation of a straight line is π‘₯π‘Ž+𝑦𝑏=1, where π‘Ž is the π‘₯-intercept of the line and 𝑏 is the 𝑦-intercept. We are, therefore, required to rearrange the equation given into the desired form.

We begin by adding 2π‘₯ to each side of the equation, giving 2π‘₯+𝑦=6.

We have now collected the π‘₯- and 𝑦-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 2π‘₯6+𝑦6=1.

Finally, we simplify the first fraction by canceling a common factor of 2 in the numerator and denominator: π‘₯3+𝑦6=1.

It is important to ensure we do not confuse the two-intercept form of the equation of a straight line with the general form π‘Žπ‘₯+𝑏𝑦=𝑐. In the previous example, we had rearranged the equation of the line to give 2π‘₯+𝑦=6 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 𝑐, we must divide every term in the equation by 𝑐 before we can use it to determine the π‘₯- and 𝑦-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 π‘₯- and 𝑦-intercepts. The π‘₯3 term reveals that the line intercepts the π‘₯-axis at (3,0), and the 𝑦6 term reveals that the line intercepts the 𝑦-axis at (0,6). As the equation of the line was originally given in slope–intercept form, we could have also identified the 𝑦-intercept from this form of the equation but the π‘₯-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 π‘₯π‘Ž+𝑦𝑏=1.

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: π‘₯π‘Ž+𝑦𝑏=1, where (π‘Ž,0) are the coordinates of the π‘₯-intercept of the line and (0,𝑏) are the coordinates of the 𝑦-intercept.

From the graph, we identify that the line intercepts the π‘₯-axis at the point (3,0) and intercepts the 𝑦-axis at the point (0,9) Hence, the values of π‘Ž and 𝑏 are 3 and 9 respectively. Substituting π‘Ž=3 and 𝑏=9 into the two-intercept form gives π‘₯3+𝑦9=1.

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 βˆ’π‘π‘Ž. We can demonstrate this another way by rearranging the equation of a straight line given in two-intercept form into the slope–intercept form 𝑦=π‘šπ‘₯+𝑏. Starting with the equation π‘₯π‘Ž+𝑦𝑏=1, we isolate 𝑦 by first subtracting π‘₯π‘Ž from each side, giving 𝑦𝑏=βˆ’π‘₯π‘Ž+1.

Multiplying both sides by 𝑏 gives 𝑦=βˆ’π‘π‘Žπ‘₯+𝑏.

This equation is now in slope–intercept form and we recall that the coefficient of π‘₯ represents the slope of the line. Therefore, we see again that the slope of the line π‘₯π‘Ž+𝑦𝑏=1 is βˆ’π‘π‘Ž. 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 π‘₯π‘Ž+𝑦𝑏=1, where (π‘Ž,0) are the coordinates of the π‘₯-intercept of the line and (0,𝑏) are the coordinates of the 𝑦-intercept, is βˆ’π‘π‘Ž.

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 𝑦-Intercepts and the Slope of a Straight Line Given in Two-Intercept Form

The graph of the equation π‘₯4+𝑦12=1 is a straight line.

  • What are the coordinates of the π‘₯-intercept of the line?
  • What are the coordinates of the 𝑦-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: π‘₯π‘Ž+𝑦𝑏=1, where (π‘Ž,0) are the coordinates of the π‘₯-intercept of the line and (0,𝑏) are the coordinates of the 𝑦-intercept. By comparing the equation of our line with the general equation above, we see that π‘Ž=4 and 𝑏=12. Hence, the coordinates of the π‘₯-intercept are (4,0) and the coordinates of the 𝑦-intercept are (0,12) .

We now recall that the slope of a line given in two-intercept form is βˆ’π‘π‘Ž. Using the values we have already determined for π‘Ž and 𝑏, the slope π‘š is equal to π‘š=βˆ’π‘π‘Ž=βˆ’124=βˆ’3.

Hence, the coordinates of the π‘₯-intercept are (4,0), the coordinates of the 𝑦-intercept are (0,12), and the slope is βˆ’3.

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 π‘₯-axis at (π‘Ž,0) and intercepts the 𝑦-axis at (0,𝑏) is π‘₯π‘Ž+𝑦𝑏=1.
  • The slope of a straight line given in two-intercept form is βˆ’π‘π‘Ž.
  • The equation of a straight line given in another form can be converted into two-intercept form by rearranging, and vice versa.

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