Which of the following represents the equation of a straight line in two-intercept form? Is it (A) 𝑥 over 𝑎 plus 𝑦 over 𝑏 equals one? (B) 𝑥 over 𝑎 plus 𝑦 over 𝑏 equals 𝑐. (C) 𝑦 equals 𝑚𝑥 plus 𝑐. (D) 𝑎𝑥 plus 𝑏𝑦 plus 𝑐 equals zero. Or (E) 𝑎𝑥 plus 𝑏𝑦 equals one.
We recall that there are many ways of writing the equation of a straight line. For example, option (D) 𝑎𝑥 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 𝑚𝑥 plus 𝑏, where 𝑚 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 𝑥- and 𝑦-axis as shown. If this line intercepts the 𝑥-axis at 𝑎 and the 𝑦-axis at 𝑏, we know that the points of intersection have coordinates 𝑎, 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 𝑥-axis at 𝑎, zero and 𝑦-axis at zero, 𝑏 is 𝑥 over 𝑎 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 𝑥 sub two minus 𝑥 sub one, where 𝑥 sub one, 𝑦 sub one and 𝑥 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 𝑥 or the rise over the run. From our diagram, we see that 𝑚 is equal to 𝑏 minus zero over zero minus 𝑎. This simplifies to 𝑏 over negative 𝑎, which can also be written as negative 𝑏 over 𝑎.
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 𝑚 multiplied by 𝑥 minus 𝑥 sub one. Substituting in the values of 𝑚, 𝑥 sub one, and 𝑦 sub one, we have 𝑦 minus zero is equal to negative 𝑏 over 𝑎 multiplied by 𝑥 minus 𝑎. Distributing the parentheses or expanding the brackets on the right-hand side, we have 𝑦 is equal to negative 𝑏 over 𝑎 𝑥 plus 𝑏. We can then divide through by 𝑏, giving us 𝑦 over 𝑏 is equal to negative 𝑥 over 𝑎 plus one. Adding 𝑥 over 𝑎 to both sides gives us the required equation 𝑥 over 𝑎 plus 𝑦 over 𝑏 is equal to one. This is the two-intercept form of the equation of a straight line which intercepts the 𝑥-axis at 𝑎, zero and the 𝑦-axis at zero, 𝑏.