Write an equation in point-slope form and general form of the line passing through (-5,3) and pe

Write an equation in point-slope form and general form of...

Key Concepts

General Form of a Line

Also known as standard form, the general form of a line is written as Ax + By + C = 0, where A, B, and C are constants. Converting an equation into general form is often required for standardizing equations, comparing lines, or applying certain algebraic techniques. It provides an alternative representation that is useful in both analysis and solving systems of equations.

Point-Slope Form

This form of the equation of a line is used when you know a specific point on the line (x?, y?) and the slope (m). It is written as y - y? = m(x - x?) and provides a straightforward way to express the line's equation directly from a known point and its slope, making it especially useful in problems where one point and a slope are given or easily computed.

Slope of Perpendicular Lines

For two lines to be perpendicular, their slopes must be negative reciprocals of each other. This means if one line has a slope m, then a line perpendicular to it will have a slope of -1/m, provided m is not zero. This concept is crucial when determining the equation of a line that must be perpendicular to an existing line, as it directly informs the correct slope to use for the new line.

Transcript

00:01 So in this question, we're going to look for the equation of the line passing through the point negative 5 .3 that's perpendicular to our line x plus 5y minus 7 equals 0.

00:11 We're going to do this in point slope form and general form.

00:16 So how are we going to do this? well, i should start by figuring out the slope of the line that i've been given, x plus 5y minus 7 equals 0, right? so what am i going to do here? well, i'm going to put this in y equals four.

00:36 So let me solve for y.

00:38 I'll subtract the x and i'll add the seven.

00:42 So 5y equals negative x plus seven, and then i'll divide by five so that i have y equals negative one -fifth x plus seven -fifths, right? and so now, now, if i want to write the point slope form of this line, i know that i'm going to have a slope of what? well, i am perpendicular to the line, y equals negative one -fifth x plus seven -fifths.

01:17 To get the slope of my perpendicular line, i have to take the opposite reciprocal of the slope of the line that i've been given.

01:28 The line that i've been given has slope negative one fifth.

01:32 And so something perpendicular to that will have a slope of positive 5.

01:39 That is the opposite reciprocal of negative 1 fifth.

01:44 And so if i wish to write the equation of our line in point slope form, well, we're going to have y minus the y coordinate of our point.

01:55 We said we're passing through negative 5 .3, so y minus 3, that's my y coordinate, is equal to my slope 5 times the quantity of x minus my x coordinate...

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