Problem 115 - Precalculus Essentials

Question

Prove that the equation of a line passing through \((a, 0)\) and \((0, b)(a \neq 0, b \neq 0)\) can be written in the form \(\frac{x}{a}+\frac{y}{b}=1\) Why is this called the intercept form of a line?

Step-by-Step Solution

Answer

The given line that passes through points (a, 0) and (0, b) can indeed be represented as x/a + y/b = 1, which is the intercept form of a linear equation. This form is so-called because the coefficients of x and y directly represent the x and y intercepts of the line, respectively.

Step 1: Finding the Equation of the Line

The first thing is to establish an equation to represent a line passing through two points. One way to do that is by using the formula for the slope of a line: \(m = (y_2 - y_1)/(x_2 - x_1)\). By substituting the coordinates of the points through which the line passes into this formula, the slope of the line can be obtained. For the given points (a, 0) and (0, b), this becomes: \(m = (0 - b)/(a - 0) = -b/a\). Then, use the slope-intercept form: \(y = mx + c\), and replace \(m\) with the value obtained and use one of the points to find \(c\). Using (a, 0): 0 = -b/a*a + c, which simplifies to c = b.

Step 2: Changing the Form to Intercept Form

Now, substitute m and c into the slope-intercept form and simplify it into the intercept form mentioned in the problem, which is written as \(x/a + y/b = 1\). The equation of the line can be written as y = -bx/a + b, which rearranges to: x/a + y/b = 1.

Step 3: Explaining the Intercept Form

The intercept form is named such because it directly shows the x and y intercepts of the line. In this form, the x-intercept is a and the y-intercept is b, which can be directly read from the equation of the line.