For nonzero values of a, b, c, and d, show that (a) lines a x+b y+c=0 and a x+b y+d=0 are parall

step 1 For non -zero values of A, B, C, and D, we want to show that A lines AX plus B -Y plus C equals

For nonzero values of a, b, c, and d, show that (a) lines a...

Key Concepts

General Form of a Line

A linear equation can be expressed in the general form Ax + By + C = 0, where A, B, and C are constants. This form is useful because it represents all straight lines in the plane, and by rearranging the equation, one can easily convert it to the slope-intercept form. It serves as the basis for analyzing many properties of lines, such as slope and position.

Slope of a Line

The slope of a line is a measure of its steepness, calculated as the ratio -A/B when the equation is written in general form Ax + By + C = 0 (with B ? 0). This concept is essential for comparing the orientations of lines, and it is instrumental in determining whether two lines are parallel or perpendicular.

Parallel Lines

Two lines are parallel if and only if their slopes are equal. When two equations differ only by a constant term but have identical coefficients for x and y, they yield the same slope upon conversion to slope-intercept form. This is why lines like Ax + By + C = 0 and Ax + By + D = 0 are parallel, since the constant adjustment does not affect their steepness.

Perpendicular Lines

Two lines are perpendicular if the product of their slopes is -1. This means that one slope is the negative reciprocal of the other. This condition guarantees that the lines intersect at a right angle, which is a key property used to prove perpendicularity in analytic geometry.

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