Question Video: Determining the Equation of a Straight Line in Two-Intercept Form Mathematics

Which of the following represents the equation of a straight line in two-intercept form? [A] (π‘₯/π‘Ž) + (𝑦/𝑏) = 1 [B] (π‘₯/π‘Ž) + (𝑦/𝑏) = 𝑐 [C] 𝑦 = π‘šπ‘₯ + 𝑐 [D] π‘Žπ‘₯ + 𝑏𝑦 + 𝑐 = 0 [E] π‘Žπ‘₯ + 𝑏𝑦 = 1

03:54

Video Transcript

Which of the following represents the equation of a straight line in two-intercept form? Is it (A) π‘₯ over π‘Ž plus 𝑦 over 𝑏 equals one? (B) π‘₯ over π‘Ž plus 𝑦 over 𝑏 equals 𝑐. (C) 𝑦 equals π‘šπ‘₯ plus 𝑐. (D) π‘Žπ‘₯ plus 𝑏𝑦 plus 𝑐 equals zero. Or (E) π‘Žπ‘₯ plus 𝑏𝑦 equals one.

We recall that there are many ways of writing the equation of a straight line. For example, option (D) π‘Žπ‘₯ plus 𝑏𝑦 plus 𝑐 equals zero is known as the general form of the equation of a straight line. As we are looking for two-intercept form, we can rule out this option. Option (C) is written in slope–intercept form. This is also sometimes written as 𝑦 equals π‘šπ‘₯ plus 𝑏, where π‘š is the slope or gradient of the line and 𝑏, or 𝑐 in this example, is the 𝑦-intercept. We can therefore also rule out this option.

Let’s consider the straight line that intercepts the π‘₯- and 𝑦-axis as shown. If this line intercepts the π‘₯-axis at π‘Ž and the 𝑦-axis at 𝑏, we know that the points of intersection have coordinates π‘Ž, zero and zero, 𝑏. We then define the two-intercept form as follows. The two-intercept form of the equation of the straight line that intercepts the π‘₯-axis at π‘Ž, zero and 𝑦-axis at zero, 𝑏 is π‘₯ over π‘Ž plus 𝑦 over 𝑏 equals one. The correct answer is option (A).

We can derive this answer as follows. We begin by recalling the slope or gradient of a line is equal to 𝑦 sub two minus 𝑦 sub one over π‘₯ sub two minus π‘₯ sub one, where π‘₯ sub one, 𝑦 sub one and π‘₯ sub two, 𝑦 sub two are two points that lie on the line. This is also sometimes referred to as the change in 𝑦 over the change in π‘₯ or the rise over the run. From our diagram, we see that π‘š is equal to 𝑏 minus zero over zero minus π‘Ž. This simplifies to 𝑏 over negative π‘Ž, which can also be written as negative 𝑏 over π‘Ž.

Next, we recall the point–slope form for the equation of a straight line. This states that 𝑦 minus 𝑦 sub one is equal to π‘š multiplied by π‘₯ minus π‘₯ sub one. Substituting in the values of π‘š, π‘₯ sub one, and 𝑦 sub one, we have 𝑦 minus zero is equal to negative 𝑏 over π‘Ž multiplied by π‘₯ minus π‘Ž. Distributing the parentheses or expanding the brackets on the right-hand side, we have 𝑦 is equal to negative 𝑏 over π‘Ž π‘₯ plus 𝑏. We can then divide through by 𝑏, giving us 𝑦 over 𝑏 is equal to negative π‘₯ over π‘Ž plus one. Adding π‘₯ over π‘Ž to both sides gives us the required equation π‘₯ over π‘Ž plus 𝑦 over 𝑏 is equal to one. This is the two-intercept form of the equation of a straight line which intercepts the π‘₯-axis at π‘Ž, zero and the 𝑦-axis at zero, 𝑏.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.