Write an equation (a) in standard form and (b) in slope-intercept form for the line described. t

step 1 This problem is looking for a line perpendicular to 3x plus 5y equals 1. That goes through the p

Write an equation (a) in standard form and (b) in...

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Key Concepts

Line Equation Forms

In analytic geometry, different forms of a line's equation are used to suit various purposes. The slope-intercept form (y = mx + b) clearly shows the slope and the y-intercept of a line, while the standard form (Ax + By = C) arranges the equation into a format that is often convenient for algebraic manipulations and solving systems of equations.

Slope and its Significance

The slope of a line represents the rate at which y changes relative to x. It is a key parameter in the slope-intercept form and is used to determine the line's steepness and direction. For instance, in the equation y = mx + b, the coefficient m is the slope of the line.

Perpendicular Lines

Two lines are perpendicular if they intersect to form a right angle. In terms of slopes, if one line has a slope m, a line perpendicular to it will have a slope of -1/m. This negative reciprocal relationship is a fundamental property used when finding the equation of a line perpendicular to a given line.

Point-Slope Form

The point-slope form of a line’s equation, given by y - y? = m(x - x?), is especially useful when you know the slope of a line and one point through which the line passes. It provides a straightforward way to form the equation before converting it into other forms such as slope-intercept or standard form.

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