Linear Equations - Algebra

Summary

This educational video script offers a comprehensive review of linear equations, focusing on three key forms: slope-intercept, standard, and point-slope. It explains how to calculate slopes using the rise-over-run method and identifies x and y-intercepts. The script also explores the concepts of parallel and perpendicular lines, their slopes, and provides step-by-step instructions for graphing linear equations in various forms, including slope-intercept, standard, and point-slope forms. It concludes with practical examples and exercises to solidify understanding.

Takeaways

• 📚 The video is a review of linear equations, aimed at helping students prepare for tests.
• 📈 There are three main forms of linear equations: slope-intercept form (y = mx + b), standard form (ax + by = c), and point-slope form (y - y1 = m(x - x1)).
• 📉 The slope-intercept form is characterized by 'm' representing the slope and 'b' representing the y-intercept.
• 🔍 In the standard form, 'a', 'b', and 'c' are coefficients, and 'x' and 'y' are variables.
• 📍 The point-slope form provides the slope (m) and a specific point (x1, y1) on the line.
• 🔢 The slope is calculated as the rise over the run, which can be positive, negative, or zero depending on the direction of the line.
• 📉 A line with a 45-degree angle has a slope of one, and the slope increases as the line becomes steeper.
• ↗️ Horizontal lines have a slope of zero, and vertical lines have an undefined slope.
• 🔍 To find the slope between two points (x1, y1) and (x2, y2), use the formula (y2 - y1) / (x2 - x1).
• 📍 X-intercepts are points where y = 0, and y-intercepts are points where x = 0.
• 🔄 Parallel lines have the same slope, and perpendicular lines have slopes that are negative reciprocals of each other.

Q & A

What are the three forms of a linear equation mentioned in the video?

The three forms of a linear equation mentioned are slope-intercept form (y = mx + b), standard form (ax + by = c), and point-slope form (y - y1 = m(x - x1)).

What does 'm' represent in the slope-intercept form of a linear equation?

In the slope-intercept form of a linear equation (y = mx + b), 'm' represents the slope of the line.

What is the slope of a line that rises 4 units and runs 3 units?

The slope of a line that rises 4 units and runs 3 units is calculated as rise over run, which is 4/3.

What is the relationship between the slopes of two parallel lines?

The slopes of two parallel lines are equal, meaning if one line has a slope of 'm', the other line will also have a slope of 'm'.

How do you find the slope of a line given two points (x1, y1) and (x2, y2)?

You can find the slope of a line given two points by using the formula: slope (m) = (y2 - y1) / (x2 - x1).

What is an x-intercept and how do you find it?

An x-intercept is a point where the line crosses the x-axis, which means the y-value is zero. To find the x-intercept, set y to zero in the equation and solve for x.

What is a y-intercept and how is it related to the 'b' value in the slope-intercept form?

A y-intercept is the y-coordinate of the point where the line crosses the y-axis, which occurs when x is zero. In the slope-intercept form (y = mx + b), 'b' represents the y-intercept.

How do you graph a linear equation in slope-intercept form?

To graph a linear equation in slope-intercept form, first plot the y-intercept, then use the slope to find another point by moving up by the rise and to the right by the run, and finally draw a line through the points.

What is the slope of a horizontal line?

The slope of a horizontal line is zero because it does not rise or fall as it moves from left to right.

What is the slope of a vertical line?

The slope of a vertical line is undefined because it represents an infinite rate of change as it moves from bottom to top without a horizontal displacement.

How do you determine if two lines are perpendicular?

Two lines are perpendicular if the product of their slopes is -1, meaning if one line has a slope of 'm', the other line will have a slope of -1/m.

What is the process of graphing a linear equation in standard form?

To graph a linear equation in standard form, find the x-intercept by setting y to zero and solving for x, find the y-intercept by setting x to zero and solving for y, plot these intercepts, and then draw a straight line through them.

How do you graph a linear equation in point-slope form?

To graph a linear equation in point-slope form, identify the point (x1, y1) and the slope 'm'. Plot the point, then use the slope to find another point by moving up by the rise and to the right by the run, and draw a line through these points.

What does it mean if a linear equation is given in the form y = a constant?

If a linear equation is given in the form y = a constant, it represents a horizontal line at the y-value of the constant, with a slope of 0.

What does it mean if a linear equation is given in the form x = a constant?

If a linear equation is given in the form x = a constant, it represents a vertical line at the x-value of the constant, with an undefined slope.

Outlines

00:00

📚 Introduction to Linear Equations

This paragraph introduces the topic of linear equations, focusing on three key forms: slope-intercept form (y = mx + b), standard form (ax + by = c), and point-slope form (y - y1 = m(x - x1)). The slope (m) is explained as the rise over run, indicating the steepness of the line, while b represents the y-intercept where the line crosses the y-axis. The paragraph sets the stage for a deeper dive into the concepts needed for understanding and solving linear equations.

05:01

📈 Understanding Slope and Intercepts

The second paragraph delves into the specifics of calculating the slope using the rise over run method and understanding the implications of positive and negative slopes on the direction of the line. It also explains the concepts of x-intercepts (points where y = 0) and y-intercepts (points where x = 0), providing examples to illustrate these concepts. The paragraph concludes with a practical example problem to identify x and y-intercepts from a set of points.

10:04

🔍 Graphing Linear Equations and Identifying Intercepts

This paragraph discusses the process of graphing linear equations, starting with identifying the slope and y-intercept from the equation. It provides a step-by-step method to plot points using the slope and then connect them to form the line. The paragraph also explains how to find x and y-intercepts from standard form equations and uses examples to demonstrate the graphing process, including identifying points where the line crosses the axes.

15:06

📐 Parallel and Perpendicular Lines

The fourth paragraph explores the relationship between parallel and perpendicular lines, explaining that parallel lines have equal slopes while perpendicular lines have slopes that are negative reciprocals of each other. It provides examples to illustrate how to determine if lines are parallel or perpendicular based on their slopes and introduces the symbols used to denote these relationships.

20:07

🤔 Practice Problems on Slopes and Intercepts

This paragraph presents practice problems that require the application of concepts learned about slopes, x-intercepts, and y-intercepts. It includes examples of calculating the slope from two given points and determining the slopes of parallel and perpendicular lines. The paragraph serves as a practical application of the theoretical knowledge discussed earlier in the script.

25:07

📉 Graphing Equations in Various Forms

The sixth paragraph focuses on the techniques for graphing linear equations in different forms, including slope-intercept, standard, and point-slope forms. It explains how to identify key components from each form and use them to plot points and draw the line. The paragraph also addresses how to graph equations that represent horizontal and vertical lines, emphasizing the undefined slope for vertical lines and zero slope for horizontal lines.

30:09

🏁 Conclusion and Practice with Multiple Choice Questions

The final paragraph wraps up the video script with a brief mention of practicing with multiple choice and free response questions to prepare for tests. It also includes an example of a multiple choice question that tests the viewer's ability to identify the correct graph for a given linear equation, emphasizing the importance of understanding slope and y-intercept.

Keywords

💡Linear Equation

A linear equation is an algebraic expression that represents a straight line when graphed on a two-dimensional plane. In the video, the main theme revolves around understanding and working with different forms of linear equations, such as slope-intercept, standard, and point-slope forms, which are essential for analyzing and graphing lines.

💡Slope-Intercept Form

Slope-intercept form is a way to write a linear equation in the format y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. The video explains this form as the starting point for understanding how to write and graph linear equations, using it to introduce the concepts of slope and y-intercept.

💡Standard Form

Standard form is another way to express a linear equation, written as Ax + By = C, where A, B, and C are constants. The video script discusses this form as a method for graphing lines, particularly by finding the x and y-intercepts to plot points on the coordinate plane.

💡Point-Slope Form

Point-slope form is a linear equation written as y - y1 = m(x - x1), where 'm' is the slope and (x1, y1) is a point on the line. The video uses this form to illustrate how knowing a point and the slope can be used to determine the equation of a line and to graph it.

💡Slope

Slope is a measure of the steepness of a line, expressed as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In the video, the concept of slope is explored in-depth, including how to calculate it using the rise over run method and its implications for the direction and steepness of a line.

💡X-Intercept

The x-intercept is the point where a line crosses the x-axis, which occurs when y = 0. The video explains how to identify x-intercepts from a set of points and how they relate to the equation of a line, particularly in the context of graphing.

💡Y-Intercept

The y-intercept is the point where a line crosses the y-axis, which is when x = 0. The video script describes the y-intercept as a key component in the slope-intercept form of a linear equation and as a point used for graphing lines.

💡Parallel Lines

Parallel lines are lines that run in the same direction and never intersect. The video script explains that parallel lines have the same slope, which is a critical concept for understanding their relationship and how to determine if lines are parallel based on their slopes.

💡Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle (90 degrees). The video discusses that the slopes of perpendicular lines are negative reciprocals of each other, providing a method to determine if lines are perpendicular by comparing their slopes.

💡Graphing

Graphing is the process of plotting the graph of an equation on a coordinate plane. The video script provides several examples of how to graph linear equations in different forms by identifying key points such as intercepts and using the slope to find additional points on the line.

Highlights

Introduction to linear equations and their forms, particularly for test preparation.

Explanation of the slope-intercept form of a linear equation: y = mx + b.

Description of the standard form of a linear equation: ax + by = c.

Introduction to the point-slope form of a linear equation: y - y1 = m(x - x1).

Discussion on the concept of slope and its calculation using the rise over run method.

Illustration of slope calculation with examples of rising and falling lines.

Explanation of the relationship between line direction and slope: positive for rising, negative for falling, zero for horizontal.

Clarification that the slope is undefined for vertical lines.

Method for calculating the slope using two points: m = (y2 - y1) / (x2 - x1).

Definition and identification of x-intercepts and y-intercepts in linear equations.

Example problem to identify x and y-intercepts from given points.

Difference between parallel and perpendicular lines and their slopes.

Explanation that parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals.

Guidance on graphing equations in slope-intercept form using the slope and y-intercept.

Method for graphing equations in standard form by finding x and y-intercepts.

Technique for graphing equations in point-slope form using a given point and slope.

How to graph horizontal and vertical lines, and understanding their slopes.

Practice problems for graphing linear equations in various forms.

Transcripts

in this video we're going to do a review

of linear equations

it's especially for those of you who

have a test that you're studying for

so

let's begin by writing down some notes

there are three forms in which you can

write a linear equation

the first one

is known as the slope

intercept form

in the slope intercept form the linear

equation is written this way

y

is equal

to mx plus b

m

represents the slope which we'll talk

about later

and b

that represents the y-intercept

we'll also

talk more about that later as well but

for now

you want to write this equation so this

is the slope-intercept form

of a linear equation

now the next form that you want to be

familiar with

is

the standard form

so to write a linear equation in

standard form

this is how it's going to look like

it's a x

plus b y

is equal to c

a b and c are simply coefficients

x and y are the variables

but when written in that form it's

called

the standard form

the next form that you need to be

familiar with is

this one the point-slope form of a

linear equation

it's y minus y one

is equal to m

times x minus x one

as the name implies this equation can

tell you the slope and the point

the slope is the value of m

so whatever number you see here

and that's the slope

the point

is x1 comma y1

now let's talk about the slope

the slope is equal to the rise

divided by the run

so let's say if you have a linear

equation

that is rising

the slope is going to be positive

and let's say you have two points on

this line

now to go from the first point

to the second point

let's say it takes you have to travel

up

by four units

this is not drawn to scale by the way

this is just an illustration so let's

say you travel up four units

and then you travel three units to the

right

so in this case your rise is four

your run is three so rise over run the

slope would be four over three

and because

it's going up the line is going up the

slope is going to be positive

now here's another example

let's say

to go from this point to that point

we need to go down

three units

so the rise is negative because we're

going down so let's say negative three

units

let's say we have a run of positive five

it's positive because we're moving to

the right

the run should always be positive

now for this one the slope is going to

be rise over run

the rise is negative 3 the run is 5 so

it's going to be negative 3 over 5.

so because

the line is going down the slope is

negative

so that's a quick and simple way to

calculate the slope

using the rise over run method

now whenever a line

it goes up at a 45 degree angle

the slope of that line is going to be

one

if it goes up like this it's about one

half

and if it goes up even steeper let's say

like this

this would be a slope of two

so this line is very steep compared to

the other ones

now if the line is horizontal

the slope

is going to be zero

if it goes down at a 45 degree angle

like this

the slope is negative one

here it's about negative a half

and here

negative two

so any time the slope increases i mean

anytime the line increases the slope is

going to be positive

if the graph is going down the slope is

negative

and for any horizontal line the slope is

zero

so if we have a line that's going to the

right or to the left the slope is zero

for a vertical line

the slope is undefined

as the line becomes more vertical the

slope increases

eventually it can go up to infinity

and at some point it will be undefined

so just know that so if you have a

vertical line the slope is undefined for

a horizontal line the slope is zero

now you can calculate the slope of a

line

if you know the two points

so let's say if the first point

is x one comma y one

and the second point

is x2 y2

the slope of the line is going to be

y2

minus y1

divided by

x2

minus x1 so here's an example let's say

the first point

is two comma five

and the second point

is let's say five

fourteen

go ahead and calculate the slope

so y2 is going to be 14

and let's replace y1 with five

x2 is five

x one is two

fourteen minus five is nine

five minus two is three

nine divided by three is three

so the slope

of the line that connects

these two points

is equal to 3.

now let's talk about

x and y-intercepts

what is an x-intercept

and what is

a y-intercept

what do you think

the answer to that question is

an x-intercept is a point

but it's a specific point

the x-intercept is the point

where y is equal to zero

so let's say you have the point three

comma zero

at this point x is three

y is zero

this particular point is an x intercept

because

the y value is zero

so the x-intercept is any value of x

when y is zero

another example of an x-intercept is the

point

negative five zero

so the x-intercept in this case will be

negative five

if we have the point

two comma zero

the x intercept is two

so any point where the y value is zero

the x value is the x intercept

for the y-intercept the situation is

reverse

the y-intercept is the y-coordinate of a

point when x is 0.

so let's say if we have the point

0

4

the y-intercept is 4.

so remember when dealing with linear

equations the y-intercept is also equal

to b

so we would say that b

is 4.

here's another example of a y-intercept

let's say the point zero negative three

so x is zero y is negative three the y

intercept

is negative three so we can say that b

is equal to negative three

now let's summarize what we've just

learned

the x intercept

is the x coordinate

of a point

that contain a y value of zero

so as we see here

this point has a y value of zero the x

intercept is the x-coordinate of that

point

so x is negative five

the y-intercept

is the y-coordinate of a point that has

an x-value of zero

so for this point

the x value is 0 but the y intercept is

the y coordinate of that point so it's y

equals 4 or b equals 4.

so that's the basics of the x and y

intercepts

so here's an example problem for you

consider

these four points the point two comma

five

negative three comma zero

one comma two

and zero comma six

given these four points identify the x

and y-intercepts

so this point here has a y value of zero

therefore that point represents the

x-intercept

the x-intercept is specifically the

x-coordinate of that point so the

x-intercept

we could say it's x equals negative

three

but it you can also say that the

x-intercept is the point negative three

comma zero you can describe it both ways

the y-intercept is the point where x is

zero so this would be the y intercept

now you can say the y intercept is y

equals six

or you could say it's a b equals six

now the next thing that we need to

review

are parallel lines

and perpendicular lines

so what's the difference between

parallel lines and perpendicular lines

what would you say

and how do their slopes

relate to each other

parallel lines

they travel in the same direction

let's call this line one and line two

let's say the slope of line one has a

value of two

if the slope of line one has a value of

two and if line two is parallel to line

one

the slope of line two will be the same

it will also be two

what you need to know is that parallel

lines they have the same slope

so m1 is going to be equal

to m2

you can also describe the relationship

between two lines that are parallel

using the symbol

if you see this

double vertical line

it means that the two lines are parallel

so this is saying l1 is parallel to l2

but

for linear equations if you have a test

we need to know is that parallel lines

they have the same slope

now let's talk about perpendicular lines

let's say we have this line

which we'll call l1

and then this line l2

perpendicular lines

they intersect at right angles that is

at 90 degrees

the slope of the perpendicular line is

the negative reciprocal of the original

line

so let's say the slope of line one

let's say it's positive

three over four

the slope of line two is going to be the

negative reciprocal so you've got to

change the sign from positive to

negative

and you've got to flip the fraction

so it's going to be negative

4 over 3.

so m1 is going to be

negative

1 over m2

so that's the relationship between the

slopes

of two perpendicular lines

so we could say l1

is perpendicular this is the symbol for

perpendicular l1 is perpendicular to l2

you can see these two lines meet at

right angles

so that's how you can describe two

perpendicular lines

so remember the slopes

are negative reciprocals of each other

now let's work on some example problems

let's say that line one

is parallel

to line two

and let's say that you're given the

slope of line one

let's say that

the slope of line one is negative three

what is the slope of line two

if they're parallel the slope of line

one is equal to the slope of line two

therefore the slope of line two will be

negative three

now let's say that line one

is perpendicular to line two

and let's say you're given the slope of

line one let's say

it's

negative four

over seven

what is the slope of line two

the slope of line two

is going to be the negative reciprocal

of the slope of line one

so the first thing you need to do is

change the sign from

negative

to positive

and then you need to flip the fraction

from four over seven to seven over four

so that's going to be the slope

of the line that's perpendicular to the

first line

now let's talk about how we can graph

equations

in slope intercept form

so let's say we have the equation y

is equal to 2x minus 4.

how can we graph this equation

so first let's put in some

marks on a graph

feel free to try this example if you

want to

so the first thing we need to identify

is the slope and the y intercept so this

is in

y equals mx plus b form it's in slope

intercept form

so we can see that the slope

is equal to two

and we can see that the y-intercept

is negative four

with this information we have everything

that we need in order to graph this

function

so here's negative four

let's go ahead and plot the y-intercept

and then from the y-intercept we can get

the second point by using the slope

so the slope is 2

which means that it's 2 over 1.

so the rise is two the run is one so to

get the next point

we're going to go

up two units

and then travel one unit to the right

so that will give us the point

one negative two so we have an x value

of one

and a y value of negative two

now let's go up two and over one again

so we get the next point

which is two comma zero so that's an

x-intercept

the x-intercept is the point

of the graph that touches the x-axis

because on the x-axis y is zero the

y-intercept

touches the y-axis

so this point

is zero negative four

it's the y-intercept because x is zero

and this is the x-intercept because y is

zero

now all you need is two points in order

to graph a line

so we can add more points but

we can just connect these points with a

straight line

so that's how we can graph

this equation in slope intercept form

now let's say we have this one y is

equal to negative three over four

x plus five

how can we graph this equation

feel free to pause the video if you want

to try it

so first let's identify the slope

and the y-intercept

so we can see that the slope

is negative 3 over 4 is the number in

front of x

the y-intercept

is 5.

so first we're gonna

plot the y-intercept

the y-intercept has the point

zero negative five x is zero y is

negative five so it's on the y axis

this is the x axis this is the y axis

so now that we have the first point the

y intercept let's use the slope to get

the next point so starting from the

y-intercept

we have a rise of negative three

and a run

of four

so this is the rise

this is the run

so as we travel down three and over four

it's going to take us to this point

so that is four on the x-axis

two on the y-axis

now all we need is two points

to graph a line so now we can just

draw a line that connects those two

points

and so that's how we can graph that

particular linear function

now let's move on to the next example

so this time we're going to graph a

linear function

in standard form

so we have 3x minus 2y is equal to 6.

so it's an ax

plus by

equals c format

how can we graph a linear equation in

standard form

what do you think we need to do

one of the most simplest techniques that

you can use is to find the x and the y

intercepts

to find the x intercept

replace y with zero

negative two times zero is zero so we

get just three x is equal to six

solving for x we can divide both sides

by three

and so we get x is equal to

six divided by three which is two

so the x intercept

is two comma zero

x is two

y is zero since we replaced y with zero

now let's find the y intercept

to find the y intercept replace x with

zero

three times zero is zero

so we're just going to get negative two

y is equal to six

dividing both sides by negative two

we get that y

is six divided by negative two which is

negative three

so the y intercept is going to be 0

negative 3.

so what we're going to do is we're going

to plot the x intercept

which is

here

it's 2 comma zero

and then let's plot the widest up

the y intercept is zero negative three

so now let's connect

these two points with a straight line

and that's all you need to do in order

to graph a linear equation in standard

form

let's try another example so let's say

we have 4x

plus 3y is equal to 12.

go ahead and graph that linear equation

so let's find the x-intercept

let's replace y

with zero

so we're going to get 4x is equal to 12

and then dividing both sides by four

we get x is 12 over four which is three

so the x intercept is three comma zero

now let's get the whiteness up

so this time let's replace x with zero

four times zero is zero

so we just get three y is equal to

twelve

divide both sides by three

so y is twelve divided by three which is

four

so we get the point zero comma four

so the x-intercept is at three

the y-intercept is at four

and then we just need to

connect those two points

with a straight line

so that's how we can graph

the linear equation in standard form 4x

plus 3y is equal to 12.

now what about an equation

that is in point slope form let's say we

have y minus 3

is equal to 2

times x minus 2.

how can we graph an equation

in that form

feel free to try that problem

so this is in y minus y1 is equal to m

times x minus x1 form

that's the point slope form in that form

we could find the point and the slope

so here's the slope the slope is 2.

now we can also find a point through

which the line passes through and that

point

is x1

y1

so what's x1 and what's y1

notice that these two negative signs are

the same

therefore

1 has to be positive 2

because those negative signs

already there

so x 1 is positive 2

y 1 is 3 without the negative sign

so when you see x minus 2 the point is

going to be 2. change the negative sign

into a positive sign

if you see y minus 3 the y coordinate is

positive 3.

so with this information we can graph it

we have a point and a slope

so let's plot the point two three

so here is two three

the x value is two the y value is three

and then we could use the slope

to get the next point

the slope is two

so we have a rise of two

and a run of one

so we can go up two and over one to get

the next point

so that's going to be three comma five

and we can go backwards

let's say if we go one to the left we

need to go down to

because there's not much space

in the right side of this graph

so that's how we can graph

a linear equation in point slope form

for the sake of practice let's do one

more example

so let's say we have the linear equation

y plus four is equal to negative three

over two

times x plus one

so go ahead and graph that linear

equation

so let's begin by identifying the slope

the slope

is negative three over two

now what's the point here we have x plus

one the x coordinate is going to be

negative one

simply reverse positive one to negative

one

here we have y plus four the y

coordinate will be negative four

so now

we have a point and a slope

that's all we need in order to graph

this function

so the first point is that negative one

negative four which is here

the x value is negative one the y value

is negative four

and then to get the next point

the slope is negative three over two

so we need to

go down three and over two but it looks

like we're out of space

so we're going to go backwards

that is we're going to go up three

and then two to the left

so up three two to the left that still

gives us the same slope

that's a rise of three a run of negative

two

which is still negative three over two

so sometimes you may need to go

backwards like in this problem

so if we go up three and over two we

should be at this point

and this point is at negative three

comma negative one

now at this point we can

go ahead and draw a line

between these two points

so that's a rough sketch

of the graph that corresponds to this

linear equation

now what would you do to graph this

equation

let's say y is equal to 3 how can you

graph that

whenever y is equal to a constant number

what you're going to get is a horizontal

line

in this case a horizontal line

at three

so if we wanted to graph y is equal to

negative two

we would simply draw a horizontal line

at negative two along the y axis

so whenever y is equal to a constant

you're going to get a horizontal line

and the slope of that line is going to

be 0.

now what if we wanted to graph x is

equal to four

in this case we're going to have a

vertical line

at x equal four

so this line will contain all points

with the x coordinate x equals four

the slope of that line

is undefined

if we want to graph x is equal to

negative 3

it's simply going to be a vertical line

touching all points

with the x coordinate negative 3.

so that's how we can graph that

now let's work on some multiple choice

and free response practice problems

that's going to help you to review for

the tests if you're studying for one

number one which of the following graphs

correspond to the equation

y is equal to two x minus three

so this equation is in slope intercept

form

now there's two things we need to focus

on

we need to identify the slope and the

y-intercept

the slope is the number in front of x

so therefore the slope

is equal to 2.

the y-intercept

is the constant that you see

next to the 2x

so the y-intercept which is b

is negative 3.

so let's identify the graph with the

correct y-intercept if we look at answer

choice a

the graph touches the y-axis at positive

three

therefore

answer choice a is not correct

looking at b c and d

the graph touches it at

negative three

so far c b and d are okay

now let's look at the slope

the first thing we want to notice is

that the slope is positive

a positive slope means that the function

is increasing

a negative slope means that it's

decreasing

and for a horizontal line the slope is

zero

so because the slope is positive the

graph should be going up

therefore we can delete d because it's

going down

graph d has a negative slope

now between b and c

what's the difference

well let's look at c

as we travel

one unit to the right

notice that the graph goes up by three

so the slope is three

now let's look at b

as we travel one unit to the right

notice that the graph goes up by two

which gives us a slope of two

so b is the right answer

it has a y-intercept of negative three

and a slope of two

you