Write the slope-intercept forms of the equations of the lines through the given point (a) parall

step 1 So we have this given line, y plus 3 is equal to 0. We want to make a line that's both parallel.

Write the slope-intercept forms of the equations of the...

Key Concepts

Perpendicular Lines

Perpendicular lines intersect at a right angle (90 degrees) and have slopes that are negative reciprocals of each other. If one line has a slope of m, then the slope of a line perpendicular to it is -1/m, provided m is not zero. This relationship is crucial when determining the equation of a line perpendicular to a given line.

Point-Slope Form

The point-slope form of a line is given by y - y? = m(x - x?), where (x?, y?) is a specific point on the line and m is the slope. This form is particularly useful when the slope of the line and one point on the line are known, allowing for a straightforward derivation of the slope-intercept form.

Parallel Lines

Parallel lines have identical slopes. This means that if two lines are parallel, the value of m in their slope-intercept forms will be the same. Given a line, any line parallel to it must have the same slope and can be expressed through the point-slope form before converting to the slope-intercept form.

Slope-Intercept Form

The slope-intercept form of a linear equation is written as y = mx + b, where m represents the slope of the line and b represents the y-intercept. This form is particularly useful for graphing linear equations and quickly identifying the rate of change and the point where the line crosses the y-axis.

Transcript

00:01 So we have this given line, y plus 3 is equal to 0.

00:03 We want to make a line that's both parallel.

00:07 We're going to do that first, and then we're going to make a line that's perpendicular to this line and goes to the point negative 1 comma 0.

00:13 So let's get this function over here in slope intercept form, so it's minus the 3 over, and we're going to see that our line is just y is equal to negative 3.

00:24 So that's going to be a horizontal line at y is equal to negative 3, parallel to the x, axis.

00:31 So for us to have a point that's going to go through this point, negative 1 comma 0, and be parallel to this point.

00:39 We need our slope to be equal to 0, since this slope here is, we could say it's y is equal to negative 3 plus 0 times x, so our slope is going to be equal to 0.

00:48 So that means that we're just going to have to have a line that goes through negative 1 comma 0 and has a slope equal to 0.

00:56 So we can say y is equal to 0, so we can say y is equal to 0 times x plus b plug in what we have for x and y so we can get zero is equal to zero times negative one plus b so zero is equal to b so our function or equation for the line that is perpendicular to y is equal to negative three and goes to the point negative one comma zero is going to be the line y is equal to zero now we're going to find a line that's perpendicular to y is equal to negative 3 and goes through the point negative 1 comma 0.

01:32 So we're going to need a line that has a negative reciprocal slope of this line.

01:39 So we see that the slope here was 0.

01:41 So if we took the negative reciprocal, that would be negative 1 divided by 0, which is an undefined slope.

01:49 The thing is that instead of doing something like that, what we can do is we can just take the line x is equal to b and then plug in what our x.

01:59 And why values are into this equation to figure out what b is equal to.

02:03 So here we'd have negative 1 is equal to b...

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