The Point-Slope Form of the Equation of a Line In Exercises 25-32, find an equation of the line

The Point-Slope Form of the Equation of a Line In Exercises...

Key Concepts

Graphing Linear Equations

Graphing linear equations involves plotting points or using properties such as slope and intercepts to draw a straight line on a coordinate plane. This process helps visualize the behavior of linear functions and confirms the relationships described by the algebraic equation.

Horizontal Line

A horizontal line is characterized by having a slope of zero, meaning there is no vertical change as the x-value varies. Its equation takes the form y = c, where c is a constant, indicating that every point on the line shares the same y-coordinate.

Point-Slope Form

This form is a method for expressing the equation of a line when you know one point on the line and its slope. It is typically written as y - y? = m(x - x?), where m represents the slope and (x?, y?) is a point on the line. This form is fundamental in understanding and deriving equations of linear functions.

Slope

The slope of a line is a measure of its steepness, indicating how much the y-value changes for a change in the x-value. It is the ratio of the vertical change (rise) to the horizontal change (run) between two points, and it determines the direction and angle of the line.

Transcript

00:01 Okay, so we're going to be taking this point and this slope and creating a line that goes through the point with the given slope.

00:07 So if we look at a line in slope intercept form, it's in the form y is equal to m times x plus b.

00:16 And so m is the slope that we are given.

00:18 So in this case, m would be negative 2 .5.

00:21 To figure out what b is going to be, we're just going to plug in the point that we have for x and y.

00:26 So we're going to plug in the y coordinate in for y and the x coordinate in for x and then find what b is equal to there and then we'll have our equation for the line through this point with the given slope in slope intercept form so we can say that negative 8 .5 is equal to negative 2 .5 multiplied by 2 .5 plus b so negative 2 .5 multiplied by 2 .5 plus b so negative 2 .5 multiplied applied by 2 .5.

00:58 I'm just going to plug that into my calculator really quick.

01:02 That is equal to 6 .25.

01:06 So we're going to have negative 8 .5 is equal to, and this is negative 6 .25 plus b.

01:14 And so now we can add 6 .25 to both sides to figure out what b is equal to.

01:20 And so we're going to get that b is equal to negative 2 .25.

01:26 And so now we have our equation in slope intercept form.

01:29 It's going to be y is equal to negative 2 .5 times x minus 2 .25.

01:37 And so now we just want to sketch an equation or sketch this line.

01:42 And so i'm just going to look at a few values of x and y.

01:49 Let's first look at the value when x is equal to zero, then y is just equal to negative 2 .25.

01:56 And when x is equal to 1, y would be equal to negative 2 .5 minus negative 2 .25, which is equal to negative 4 .75.

02:07 And then let's look at y is equal to negative 1.

02:10 So that would be negative 1 times negative 2 .5, which is equal to positive 2 .5 minus 2 .25.

02:18 Would be equal to just 0 .25...

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