is it just me or is this lesson hard to understand than the others

to be honest with you......i am confused

hey, they make a mistake on the last question.4 times -1 suppose equal to -4, not 4

There is, in fact, a typo in the second line, but I'm pretty sure that doesn't lead to a mistake, as you're claiming (and I believe that the typo added extra confusion in your mind). They should either have just flipped the inequality sign and multiplied one side by (-1) or have left the flipping of the inequality from "greater than or equal to" to "less than or equal to" to be done in the following line. However, they have omitted the multiplication of negative one on the right-hand side from line three.

The second line has a slope of 6 and passes through the point (−1, −9). This means the line also passes through the point (-1 + 1, -9 + 6), or (0, - 3). Its equation is y = 6x - 3 Please someone explain this to me. Why does a line that has a slope of 6 and passes through (−1, −9) also pass through point 1 and 6?

We're given a slope and a point on the line (-1,-9) Now, we are aware that our slope is ∆y/∆x. That means, our slope is 6/1 which shows that for every 1 we move in x, we move up 6 in y. That also means if we change x by 1,we change y by 6. Hence, (-1+1,-9+6) The original points were (-1,-9) We want to change x by 1 and y by 6 to get our y-intercept. (x+1,y+6) I hope this helps

I couldn't understand the practice: intersection of two lines

dear,after drawing the two lines on the graph just look for where they two hit together that's the intersects of the lines.

If x-y<z, then is y<x-z or y>x-z correct? Thank you!

Whenever we work with inequalities and multiply/divide by a negative number, we have to flip the inequality sign: x - y < z -y < z - x y > -z + x y > x - z As for why, this is a little bit of a logic step. If we multiply/divide both sides by a negative, the thing that was larger before is still larger in magnitude, but because of the negative that means that its value is actually smaller. This means we need to reverse our inequality sign.

Main content

Course: Digital SAT Math > Unit 2

Lesson 8: Graphs of linear systems and inequalities: foundations

Graphs of linear systems and inequalities | Lesson

Google Classroom

A guide to graphs of linear systems and inequalities on the digital SAT

What are graphs of linear systems and inequalities problems?

Graphs of linear systems and inequality problems deal with graphs of the following in the

x y

-plane:

In this lesson, we'll learn to:

Identify solutions to linear systems and inequalities
Match graphs with systems of linear equations and inequalities

This lesson builds upon an understanding of the following skills:

Solving linear equations and linear inequalities
Graphs of linear equations and functions
Solving systems of linear equations

You can learn anything. Let's do this!

How are systems of linear equations related to their graphs?

Systems of equations with graphing

Khan Academy video wrapper

Systems of equations with graphingSee video transcript

Intersection points as solutions

In a system of linear equations, each equation is represented by a line in the

x y

-plane. The intersection of these lines represents the solution to the system.

For example, the graphs of

y = x

and

y = - x

intersect at

(0, 0)

, which is the solution to the system of equations.

Try it!

Try: identify the solution of a system of equations by its graph

In the graph above, the line with the positive slope has a slope of

1

and a

y

-intercept of

. The line with the negative slope has a slope of

and a

y

-intercept of

1

The two lines intersect at

(

), which is the solution

(x, y)

to the following system of equations

y = x - 2

and

y = - 2 x + 1

How do I graph linear inequalities?

Intro to graphing two-variable inequalities

Khan Academy video wrapper

Intro to graphing two-variable inequalitiesSee video transcript

Representing solutions with shaded areas

When we learned to solve linear inequalities, we learned that multiple values can satisfy an inequality. How do we graph the infinite number of

(x, y)

values that can satisfy a linear inequality?

We do it by shading the

x y

-plane! A linear inequality is defined by a line in the

x y

-plane. The line divides the plane into two halves, and which half of the plane we shade depends on the inequality sign.

It's the easier to determine which half of the plane to shade when the inequality is in slope-intercept form. When we have linear inequalities in slope-intercept form:

If $y$ ‍ is greater than $m x + b$ ‍, shade above the line.
If $y$ ‍ is less than $m x + b$ ‍, shade below the line.

But what about points on the line? In slope-intercept form, inequalities with...

Greater than ( $>$ ‍) or less than ( $<$ ‍) signs do not include points on the line in the solution set. We use a dashed line to show that the points on the line are not included.
Greater than or equal to ( $\geq$ ‍) or less than or equal to ( $\leq$ ‍) signs do include points on the line in the solution set. We use a solid line to show that points on the line are included.

For example, for

y > x

, we draw

y = x

as a dashed line and shade above the line:

Try it!

Try: verify solutions to a linear inequality

The inequality

y > - \frac{1}{3} x - 2

is graphed in the

x y

-plane above.

The point

(1, 1)

the shaded area, so it

a solution to the inequality.

The point

(- 3, - 4)

the shaded area, so it

a solution to the inequality.

The point

(0, - 2)

is on the line

y = - \frac{1}{3} x - 2

. It

a solution to the inequality because the

>

sign indicates that points on the line

part of the solution set.

How do I graph systems of linear inequalities?

Intro to graphing systems of inequalities

Khan Academy video wrapper

Intro to graphing systems of inequalitiesSee video transcript

How do I identify the region representing a system of linear inequalities?

Two intersecting lines will always divide the

x y

-plane into four regions. Points in each of the four regions represent solutions to a different system of linear inequalities.

In the graph below, the equations of the lines defining the four regions are:

\begin{aligned} y & = 3 x - 2 \\ y & = - \frac{1}{3} x + \frac{4}{3} \end{aligned}

Replacing the equal signs with inequality signs lets us specify one of the four regions.

When we have linear inequalities in slope-intercept form:

If $y$ ‍ is greater than $m x + b$ ‍, shade above the line.
If $y$ ‍ is less than $m x + b$ ‍, shade below the line.

For example, let's look at the following system:

\begin{aligned} y & \leq 3 x - 2 \\ y & \geq - \frac{1}{3} x + \frac{4}{3} \end{aligned}

This means the solutions to the system of linear inequalities are represented by the region below the line

y = 3 x - 2

and above the line

y = \frac{1}{3} x + \frac{4}{3}

Try it!

TRY: DETERMINE THE INEQUALITY SIGNS FROM A GRAPH

The the graph above represents a system of linear inequalities.

Because the region

the line

y = - \frac{1}{3} x + \frac{4}{3}

is shaded,

y

or equal to

- \frac{1}{3} x + \frac{4}{3}

Because the region

the line

y = 3 x - 2

is shaded,

y

or equal to

3 x - 2

Your turn!

practice: find the intersection of two lines

The graph of a line in the

x y

-plane has a slope of

3

and passes through the origin. The graph of a second line has a slope of

6

and passes through the point

(- 1, - 9)

. If the two lines intersect at the point

(a, b)

, what is the value of

a

practice: match a linear inequality to its graph

If the shaded region in the graph above represents the solution set to an inequality, which of the following could be the inequality?

Practice: match a system of linear inequalities to its graph

\begin{aligned} x - y \geq 4 \\ y \leq - \frac{3}{4} x - \frac{1}{2} \end{aligned}

In which of the following does the shaded region represent the solution set in the

x y

-plane to the system of inequalities above?

Things to remember

When a system of linear equations is graphed, the solution appears where the lines intersect.

When we have linear inequalities in slope-intercept form:

If $y$ ‍ is greater than $m x + b$ ‍, shade above the line.
If $y$ ‍ is less than $m x + b$ ‍, shade below the line.

In slope-intercept form, inequalities with...

Greater than ( $>$ ‍) or less than ( $<$ ‍) signs do not include points on the line in the solution set. We use a dashed line to show that the points on the line are not included.
Greater than or equal to ( $\geq$ ‍) or less than or equal to ( $\leq$ ‍) signs do include points on the line in the solution set. We use a solid line to show that points on the line are included.

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anungoo0531
Posted a year ago. Direct link to anungoo0531's post “is it just me or is this ...”
is it just me or is this lesson hard to understand than the others
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- okwor.amarachi
  Posted 6 months ago. Direct link to okwor.amarachi's post “to be honest with you.......”
  to be honest with you......i am confused
  Comment on okwor.amarachi's post “to be honest with you.......”
  (29 votes)
wobuzhidao.1206
Posted 9 months ago. Direct link to wobuzhidao.1206's post “hey, they make a mistake ...”
hey, they make a mistake on the last question.4 times -1 suppose equal to -4, not 4
Button navigates to signup pageComment on wobuzhidao.1206's post “hey, they make a mistake ...”
(35 votes)
Answer
- Anup Aryal
  Posted 4 months ago. Direct link to Anup Aryal's post “There is, in fact, a typo...”
  There is, in fact, a typo in the second line, but I'm pretty sure that doesn't lead to a mistake, as you're claiming (and I believe that the typo added extra confusion in your mind). They should either have just flipped the inequality sign and multiplied one side by (-1) or have left the flipping of the inequality from "greater than or equal to" to "less than or equal to" to be done in the following line. However, they have omitted the multiplication of negative one on the right-hand side from line three.
  Comment on Anup Aryal's post “There is, in fact, a typo...”
  (7 votes)
s ss
Posted a year ago. Direct link to s ss's post “why i can't understand th...”
why i can't understand this lesson , i have watched the videos 3 times
Button navigates to signup pageComment on s ss's post “why i can't understand th...”
(28 votes)
Answer
tetzjord
Posted 7 months ago. Direct link to tetzjord's post “Tip: if you're solving fo...”
Tip: if you're solving for y in an inequality and move it to the other side, make sure to pay attention that the inequality sign has "switched" for the shaded region.
Button navigates to signup pageComment on tetzjord's post “Tip: if you're solving fo...”
(23 votes)
Answer
Jon
Posted a year ago. Direct link to Jon's post “The second line has a slo...”
The second line has a slope of 6 and passes through the point (−1, −9). This means the line also passes through the point (-1 + 1, -9 + 6), or (0, - 3). Its equation is y = 6x - 3

Please someone explain this to me. Why does a line that has a slope of 6 and passes through (−1, −9) also pass through point 1 and 6?
Button navigates to signup pageButton navigates to signup page
(13 votes)
Answer
- Emmanuel David
  Posted a year ago. Direct link to Emmanuel David's post “We're given a slope and a...”
  We're given a slope and a point on the line (-1,-9)
  Now, we are aware that our slope is ∆y/∆x. That means, our slope is 6/1 which shows that for every 1 we move in x, we move up 6 in y. That also means if we change x by 1,we change y by 6.
  Hence, (-1+1,-9+6)
  The original points were (-1,-9)
  We want to change x by 1 and y by 6 to get our y-intercept. (x+1,y+6)
  
  I hope this helps
  Comment on Emmanuel David's post “We're given a slope and a...”
  (15 votes)
amina
Posted 8 months ago. Direct link to amina's post “Guys if you are strugglin...”
Guys if you are struggling with the graphs go and check out the betterthanyourprof youtube channel the guy explained it in the easiest way possible
Button navigates to signup pageComment on amina's post “Guys if you are strugglin...”
(15 votes)
Answer
kama emae
Posted a year ago. Direct link to kama emae's post “mistake in the last pract...”
mistake in the last practice, x−y≥4, after isolating y in will be y=<x-4
Button navigates to signup pageComment on kama emae's post “mistake in the last pract...”
(12 votes)
Answer
Niffy
Posted 10 months ago. Direct link to Niffy's post “I am struggling with most...”
I am struggling with most of these questions, I am not really good when it comes to graphs. Any suggestions pls
Button navigates to signup pageComment on Niffy's post “I am struggling with most...”
(9 votes)
Answer
- sohail karim
  Posted 9 months ago. Direct link to sohail karim's post “practice is the only way”
  practice is the only way
  Button navigates to signup page
  (6 votes)
azkakhan909
Posted a year ago. Direct link to azkakhan909's post “I couldn't understand the...”
I couldn't understand the practice: intersection of two lines
Button navigates to signup pageComment on azkakhan909's post “I couldn't understand the...”
(7 votes)
Answer
- Phina Stephanie
  Posted a year ago. Direct link to Phina Stephanie's post “dear,after drawing the t...”
  dear,after drawing the two lines on the graph just look for where they two hit together that's the intersects of the lines.
  Button navigates to signup page
  (7 votes)
LE KHOI NGUYEN
Posted a year ago. Direct link to LE KHOI NGUYEN's post “If x-y<z, then is y<x-z o...”
If x-y<z, then is y<x-z or y>x-z correct? Thank you!
Button navigates to signup pageComment on LE KHOI NGUYEN's post “If x-y<z, then is y<x-z o...”
(4 votes)
Answer
- Hecretary Bird
  Posted a year ago. Direct link to Hecretary Bird's post “Whenever we work with ine...”
  Whenever we work with inequalities and multiply/divide by a negative number, we have to flip the inequality sign:
  x - y < z
  -y < z - x
  y > -z + x
  y > x - z
  As for why, this is a little bit of a logic step. If we multiply/divide both sides by a negative, the thing that was larger before is still larger in magnitude, but because of the negative that means that its value is actually smaller. This means we need to reverse our inequality sign.
  Comment on Hecretary Bird's post “Whenever we work with ine...”
  (13 votes)