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Chapter 2: Problem 115

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?

Short Answer

Expert verified
The intercept form of the equation is \(\frac{x}{a} + \frac{y}{b} = 1\). It is called 'intercept form' because the inverse of the coefficients of \(x\) and \(y\) give the \(x\) and \(y\) intercepts respectively.

Step by step solution

01

Establish the Points

It is specified that the line passes through the points (a, 0) and (0, b). These are the x and y-intercepts of the line respectively.
02

Derive the Equation Using Point-Slope Form

The point-slope form of a line with slope m passing through the point (x1, y1) is given by \(y - y1 = m (x - x1)\). Considering the slope as \(m = -b/a\), and using the point (a, 0), the equation becomes \(y - 0 = -b/a (x - a)\) simplifying this will give \(ax + by = ab\).
03

Convert to Intercept Form

Divide the whole equation by \(ab\) to obtain the intercept form. Thus, \(\frac{x}{a} + \frac{y}{b} = 1\)is the required equation of the line.
04

Understanding Intercept Form

This form is called the intercept form of a line because the inverse of the coefficient of x and y gives the x and y intercepts of the line respectively. In this case, \(a\) is the x-intercept and \(b\) is the y-intercept of the line.

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