Find an equation of each line with the given slope that passes through the given point. Write th

Find an equation of each line with the given slope that...

Key Concepts

Slope

Slope represents the rate of change of a line, indicating how much the vertical coordinate changes for a unit change in the horizontal coordinate. It is a fundamental concept in linear equations and calculus, used to describe the steepness and direction of a line.

Point-Slope Form

Point-slope form is a way to write the equation of a line when a point on the line and its slope are known. This form is typically expressed as y - y? = m(x - x?) and is very useful for quickly constructing a linear equation from given data.

Standard Form of a Linear Equation

The standard form is a particular way of writing a linear equation, usually in the format Ax + By = C where A, B, and C are integers. Writing the line in this form can simplify further analysis and is often required in many applications and examinations.

Algebraic Manipulation and Conversion

Algebraic manipulation involves the processes of substitution, clearing fractions, and rearranging terms to convert one form of a linear equation to another. This skill is essential for moving from the point-slope form to the standard form or any other desired representation.

Transcript

00:01 All right, this equation is asking me to write the equation of the line, and they're giving me the slope.

00:09 So i'm going to go ahead and label this my slope, my m, and they're giving me a point.

00:15 When they give me a point, they're actually giving me an x value and a y value.

00:19 But we do have to write this in standard form.

00:23 So this is my standard form right over here.

00:25 Well, in order to get that, i do need to first use my slope intercept form.

00:33 Because if i have this in slope intercept form, then i could easily move this value over to the other side.

00:41 And that would help me get it into standard form.

00:44 And i do notice i have my m, but i don't have my b.

00:49 But that's okay because i have some other stuff here.

00:52 So let's plug in what we have.

00:53 To begin with, i have my m is 3 over 2.

00:58 So let's plug that in for the m.

01:00 I'm going to go ahead and put that in parentheses just because he's a fraction, so that can make things a little bit more challenging.

01:07 I also notice i have an x value.

01:09 So right here next to it, i'm going to put a 5.

01:13 And i might even want to put the 5 over 1 since i am dealing with a fraction there.

01:18 All right.

01:19 I also notice i have a y value, which is negative 6.

01:22 The only thing i don't have is my b, which now looks like i have enough information to solve for b.

01:31 So let's begin.

01:33 Since i went ahead and put that 5 over 1, i can multiply these two fractions pretty easily.

01:39 I can multiply the numerators, 3 times 5 is just 15, and i can multiply the denominators, 2 times 1 is just 2.

01:48 I still have a plus b, and i still have a negative 6.

01:53 Since this value is a fraction, i'm now going to go ahead and make this value a fraction because i know now i'm going to have to move this 15 over 2 to the other side, which means i have to subtract 15 over 2 from both sides of my equation.

02:12 But the good news is, is when that is done, i will know what b equals.

02:17 So i'm going to take a little bit off to the side here because i want to make sure i do this correctly.

02:23 I have negative 6 over 1, and i am subtracting 15 over 2.

02:30 Well, in order to subtract fractions, we need a common denominator.

02:34 And i notice the common denominator here would be a 2.

02:38 So i'm going to multiply the top and the bottom of this first fraction by 2.

02:43 That will give me a negative 12 over 2 minus 15 over 2, a negative 12 minus 15 over 2...

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