Solve a Linear System by Graphing | jensenmath.ca | grade 10

Summary

This video script introduces the concept of solving linear systems in a grade 10 math course. It defines a linear system as two or more linear equations considered together and explains the importance of finding the point of intersection. The script outlines three methods to solve these systems: graphing, substitution, and elimination, focusing on the graphing method in this lesson. It demonstrates how to rearrange equations into slope-intercept form, graph the lines, and find their intersection points. The video also discusses the possible outcomes of solving linear systems, including one solution, no solution, or infinite solutions, and emphasizes the importance of verifying solutions.

Takeaways

• 📚 The lesson is focused on solving linear systems, which are sets of two or more linear equations considered simultaneously.
• 📈 A linear system is solved by finding the point of intersection where two or more lines cross, graphically represented as the values of x and y that satisfy all equations.
• 📊 There are three main methods for solving linear systems: graphing, substitution, and elimination, with the first lesson covering graphing.
• 📉 To graph lines, equations are rearranged into the slope-intercept form \( y = mx + b \), where m is the slope and b is the y-intercept.
• 📐 The slope of a line is calculated using the formula \( \frac{change\ in\ y}{change\ in\ x} \) or \( \frac{y2 - y1}{x2 - x1} \).
• 🔍 When graphing, one can use the slope and y-intercept, x and y intercepts, or a table of values to plot the lines accurately.
• 🤔 There are three possible outcomes when solving a linear system: one solution (lines intersect), no solution (parallel and distinct lines), or infinitely many solutions (parallel and coincident lines).
• 📝 The process of solving by graphing involves plotting both lines, finding their point of intersection, and verifying the solution by substituting the intersection point into the original equations.
• 🔢 The script provides step-by-step examples of graphing lines and finding their points of intersection, emphasizing the importance of clear communication and verification of the solution.
• 📉 The importance of verifying the solution is highlighted to ensure the intersection point satisfies both equations in the system, confirming its accuracy.
• 🚫 The script also discusses scenarios where the lines do not intersect, such as when they are parallel and distinct, resulting in no solution, or when they are parallel and coincident, resulting in infinitely many solutions.

Q & A

What is the main focus of the first unit of the grade 10 math course?

The main focus of the first unit of the grade 10 math course is solving linear systems.

What is a linear system in mathematics?

A linear system is a set of two or more linear equations that are considered at the same time.

What is the definition of the point of intersection in the context of linear systems?

The point of intersection is the point where two or more lines cross, which is the solution to the linear system when solved graphically.

What are the three main methods for solving a linear system mentioned in the script?

The three main methods for solving a linear system are graphing, substitution, and elimination.

How does the script define 'solving' a linear system?

Solving a linear system means finding the values of the variables that satisfy all of the equations in the system.

What is the equation in the form of y equals mx plus b used for?

The equation y equals mx plus b is used to describe the relationship between the x and y coordinates of any point on a line segment in terms of the slope (m) and the y-intercept (b).

How can the slope of a line be calculated according to the script?

The slope of a line can be calculated using the slope formula, which is the change in y over the change in x, written as (y2 - y1) / (x2 - x1).

What are the possible outcomes when solving a linear system graphically?

The possible outcomes are one solution (if the lines intersect at one point), no solution (if the lines are parallel and distinct), or infinitely many solutions (if the lines are parallel and coincident).

Why is it important to verify the solution after finding the point of intersection?

It is important to verify the solution by plugging the x and y values of the point of intersection into both original equations to ensure that the point satisfies both equations, confirming it is the correct solution to the linear system.

How does the script describe the process of graphing lines using the slope and y-intercept method?

The script describes the process as rearranging the linear equations into the form y equals mx plus b, then using the slope (m) and y-intercept (b) to plot the y-intercept and additional points using the slope to fill the grid, and finally connecting the points with a straight line.

What is the significance of the y-intercept in the context of graphing lines?

The y-intercept is the point where the line crosses the y-axis, and it is used in conjunction with the slope to accurately graph the line.

How can you determine if two lines are parallel?

Two lines are parallel if they have the same slope but different y-intercepts.

What does it mean if two lines are coincident?

If two lines are coincident, it means they have the same slope and y-intercept, and therefore, they lie exactly on top of each other, indicating an infinite number of solutions to the linear system.

How does the script suggest communicating the solution to a linear system?

The script suggests communicating the solution clearly, either by stating the point of intersection as an (x, y) coordinate or by stating the values of x and y separately, and ensuring the solution is well-organized and easy for the teacher to identify.

Outlines

00:00

📚 Introduction to Solving Linear Systems

This paragraph introduces the first lesson of the grade 10 math course, focusing on solving linear systems. It defines a linear system as two or more linear equations considered simultaneously and explains the point of intersection where two or more lines cross. The objective is to find the values of variables that satisfy all equations in the system. The three main methods for solving linear systems—graphing, substitution, and elimination—are briefly introduced, with a focus on solving by graphing in this lesson.

05:01

📊 Graphing Lines Using Slope and Y-Intercept

This paragraph explains the method of graphing lines using the slope and y-intercept. It details how to rearrange linear equations into the form y = mx + b, where m is the slope and b is the y-intercept. The paragraph covers how to graph lines by plotting the y-intercept and using the slope to plot additional points. It also discusses the different scenarios when lines intersect: at one point (different slopes), no points (parallel and distinct), or infinitely many points (parallel and coincident).

10:02

📐 Example: Graphing Two Linear Equations

This paragraph walks through an example of graphing two linear equations: y = x + 4 and y = -x + 2. It explains the process of identifying the slope and y-intercept for each equation, plotting the y-intercepts, and using the slopes to plot additional points. The point of intersection for the two lines is identified and verified by substituting the intersection point into both equations to ensure it satisfies both. The importance of clearly communicating the final solution is emphasized.

15:04

📈 Verifying Solutions and Solving Another Example

The paragraph continues with verifying solutions by checking if the intersection point satisfies both equations. It then presents another example with two different lines: 2x + y = 5 and x - 2y = 10. The process of rearranging equations into slope-intercept form, plotting the y-intercepts, and using the slopes to graph the lines is detailed. The intersection point is identified and verified, demonstrating the method's accuracy.

20:04

📏 Estimating Intersection Points When Graphing

This paragraph discusses the challenges of estimating intersection points when graphing. It provides an example with two lines: 2x + 5y = -20 and 5x - 3y = -15. The equations are rearranged into slope-intercept form, and the lines are graphed. The intersection point is estimated due to the graph's scale limitations. The importance of learning algebraic methods in future lessons to avoid estimation errors is highlighted.

🔀 Exploring Parallel and Coincident Lines

This paragraph explores scenarios where lines are parallel and either distinct or coincident. It provides examples with equations that have the same slope but different y-intercepts (parallel and distinct, resulting in no solutions) and equations with the same slope and y-intercepts (parallel and coincident, resulting in infinitely many solutions). The graphical representation of these scenarios is explained to illustrate why these cases result in different numbers of solutions.

📏 Conclusion and Transition to Algebraic Methods

The final paragraph summarizes the lesson on solving linear systems by graphing, reiterating the different possible outcomes (one solution, no solutions, infinitely many solutions). It emphasizes the need for algebraic methods like substitution and elimination, which will be covered in upcoming lessons, to provide precise solutions without relying on graphical estimation. The lesson ends with a preview of the next topic: solving linear systems using the method of substitution.

Keywords

💡Linear System

A linear system refers to a collection of two or more linear equations that are considered simultaneously. In the context of the video, the main theme revolves around solving these systems to find the values of variables that satisfy all equations involved. The script explains that solving a linear system graphically involves finding the point of intersection where the lines represented by the equations meet.

💡Point of Intersection

The point of intersection is the exact location where two or more lines cross each other on a graph. The video emphasizes that finding this point is crucial when solving linear systems graphically, as it represents the solution to the system where all equations are satisfied simultaneously.

💡Graphing

Graphing is a method used to visually represent and solve linear systems by plotting lines on a coordinate plane. The script describes the process of graphing lines using their slope and y-intercept to find the point where they intersect, which is the solution to the system.

💡Slope

Slope is a fundamental concept in the script that defines the steepness or incline of a line. It is calculated as the change in y over the change in x, often represented as 'rise over run.' The video uses slope to graph lines and determine their position relative to each other on the coordinate plane.

💡Y-Intercept

The y-intercept is the point where a line crosses the y-axis on a graph. In the script, the y-intercept is used alongside the slope to graph lines and find their equation in the form y = mx + b, which is essential for solving linear systems.

💡Substitution

Although not explicitly detailed in the script, substitution is mentioned as one of the algebraic methods for solving linear systems. It involves solving one equation for one variable and then substituting the result into the other equation to find the solution.

💡Elimination

Like substitution, elimination is an algebraic method hinted at in the script for solving linear systems. It involves adding or subtracting equations to eliminate one variable, making it possible to solve for the other variable.

💡Parallel Lines

The script discusses parallel lines, which are lines with the same slope but different y-intercepts. These lines never intersect and represent a scenario in linear systems where there are no solutions because they do not meet at any point.

💡Coincident Lines

Coincident lines are lines that overlap exactly, having the same slope and y-intercept. The script explains that when lines are coincident, there are infinitely many solutions to the linear system because every point on one line is also on the other.

💡Verification

Verification in the context of the script means checking the solution to ensure it satisfies all equations in the linear system. The video demonstrates how to plug the solution back into the original equations to confirm that it is indeed the correct point of intersection.

💡Graphical Solution

A graphical solution is the result obtained by visually representing and solving linear systems on a graph. The script emphasizes the importance of graphical solutions as a method to find the point of intersection of lines, which corresponds to the solution of the system.

Highlights

Introduction to the concept of a linear system and its definition as two or more linear equations considered simultaneously.

Explanation of the point of intersection as the point where two or more lines cross.

Objective of solving a linear system is to find values of variables that satisfy all equations in the system.

Graphical representation of solving linear systems by finding the point where two lines intersect.

Introduction of three main methods for solving linear systems: graphing, substitution, and elimination.

Emphasis on the method of graphing lines using the slope and y-intercept for solving linear systems in this lesson.

Description of rearranging linear equations into the form y = mx + b for graphing purposes.

Explanation of the slope formula and its calculation using the change in y over change in x.

Different methods for graphing lines, including using x and y intercepts or creating a table of values.

Discussion of three possibilities when solving linear systems: one solution, no solutions, or infinitely many solutions.

Visual demonstration of the three possibilities for solving linear systems with different slopes.

Steps for solving linear systems by graphing, including rearranging equations and verifying the solution.

Example of solving a linear system with two lines, y = x + 4 and y = -x + 2, using graphing.

Verification process of a solution by plugging the point of intersection back into the original equations.

Demonstration of solving a second linear system with lines 2x + y = 5 and x - 2y = 10.

Illustration of estimating the point of intersection when exact points are not easily identifiable on the graph.

Examples of linear systems with no solutions due to parallel and distinct lines.

Examples of linear systems with infinitely many solutions due to parallel and coincident lines.

Conclusion and preview of the next lesson on the method of substitution for solving linear systems.

Transcripts

[Music]

let's get started with lesson one of the

first unit of the grade 10 math course

in this unit it's going to be focused on

solving linear systems so let's get that

definition out of the way first what is

a linear system a linear system is two

or more linear equations that are

considered at the same time

another definition we'll need is the

point of intersection

the point of intersection is the point

where two or more lines cross and like i

said in this unit we're focusing on

solving linear systems so what does

solving mean to solve a linear system

means to find the values of the

variables that satisfy all of the

equations in the system so we're going

to be finding the values of x and y

that satisfy both equations

graphically speaking that means we're

finding the point x y where the two

lines will intersect now there are three

main methods for solving a linear system

you can solve by graphing or there are

two algebraic ways called substitution

and elimination that we'll cover in

lessons two and three in this lesson

we're just going to be solving by

graphing and to solve by graphing we

basically just have to graph both lines

in the system and see where their point

of intersection is now there are lots of

ways to graph lines i'm going to be

using method one for graphing lines

which is using the slope and y-intercept

to graph the lines and to do that you'd

have to first rearrange the linear

equations into the form y equals mx plus

b then you can use m the slope and b the

y intercept to accurately graph the line

y equals mx plus b

is an equation that describes the

relationship between the x and y

coordinates of any point on a line

segment

it describes the relationship in terms

of m

the slope of the line which we can

calculate by doing the slope formula

change in y over change in x which we

could write as y2 minus y1 over x2 minus

x1 and b

the y-intercept of the line

you don't always have to use that method

you're welcome to graph it using its x

and y intercepts or creating a table of

values for the equation any method you

use should get you an accurate graph of

the line and allow you to find the point

of intersection let me go over the three

possibilities for what could happen when

you are trying to solve a linear system

before we fill out this chart let me

give you a visual demonstration of this

if the lines are not parallel meaning

they have different slopes the lines

will intersect at one point meaning we

will get one solution for x and y

if the lines are parallel meaning they

have the same slope they're either going

to be parallel and distinct meaning

they're going to have no solutions or

they'll be parallel and coincident

meaning that they have an infinite

number of solutions

so in this table let's fill out that

information so if the lines intersect at

one point we know that the slopes are

different

comparing the x and y intercepts of two

non-parallel lines isn't necessary to

determine if it has a point of

intersection or not we know if the lines

aren't parallel they have a point of

intersection

the y-intercepts and the x-intercepts

are usually different from each other

between the lines but i suppose they

could be the same if

on the x or y-axis is where the point of

intersection is

so i'll say that the intercepts are

usually different unless the lines

intersect on an axis how many solutions

do we get if we have different slopes

we get

one solution one point of intersection

now lines are parallel and distinct if

they have

the same slope

but

different

intercepts

if that happens we get no solutions to

the linear system

and you can see graphically there's

going to be no point where those lines

intersect

if the lines are parallel and coincident

it means the lines are right on top of

each other and that's because they have

the exact same slope and the exact same

intercepts

meaning we get infinitely many solutions

every point that's on one line is also

on the other line that's why there's

infinitely many solutions now we're

going to solve five linear systems

together using the method of graphing

here are the steps we're going to follow

for each of those systems we're going to

start by rearranging each linear

equation into the form y equals mx plus

b

i'll then graph both of the equations

and find the point of intersection

once i figure out where the point of

intersection is it's important that we

verify our answer plug the xy point of

the point of intersection into both of

the original equations to make sure that

xy point satisfies both equations that

verifies it's on both lines and proves

you have the correct answer and we

should clearly communicate our solution

making sure that your teacher is going

to be able to see what you think your

final answer is

let's try example one i have a linear

system i have two lines y equals x plus

four and y equals negative x plus two

i'm just going to show some rough work

for both lines and i'll do this for each

of the examples and i'll color code it

so i have

line one n is y equals one x plus four

now usually we don't write a coefficient

of one but i'm going to put it there

just so you can see that that is the

slope of this line right a line that's

in the form y equals mx plus b

its slope is equal to the m value which

in this case i see m is right here which

is 1. now when graphing lines it helps

to think of the slope as a fraction

right because slope is rise over run so

any whole number you could rewrite as

over 1.

so what i have here here's my m value

and my slope remember is rise

over run

and also i'm going to need to point out

what the y intercept is the y intercept

is the b value of the equation

the b value is 4

so i'm going to write the y intercept is

equal to the b value

which is 4. now to graph this line all i

have to do is first plot the y intercept

of 4

and then use my slope of 1 over 1 right

rise 1 run 1 to plot more points to fill

this grid so from my y-intercept i'm

going to rise one and run one remember a

positive rise means up and a positive

run means right so up one right one i'm

going to fill this grid

and now to plot points on the other side

of the y-intercept we do the exact

opposite instead of up one right one we

go down one left one

and notice these points also fall on the

same line

now i'm going to connect these points

with a straight line

and label this as line one

make sure when you're graphing lines you

put arrows on both ends of the line

showing that it does continue

next let me do the same process but for

line two line two is y equals negative

one x

plus 2. now notice this is also in the

format y equals mx plus b already

where

the slope

is equal to the m value which in this

case is negative 1 and like i said any

whole number you could rewrite as over 1

so that makes it easier to graph

and the y-intercept is the b value

which with this equation is 2.

here's our m-value it's the coefficient

of x here's our b value it's the

constant added after the x

and my slope i wrote it as a fraction

because slope is rise

over run so to graph this line we always

start by plotting the y intercept of 2

and then do rise over run

to plot points to fill the grid

so rise of negative one means down one

and run one means right one so i go down

one right one

and then to plot points on the other

side of the y-intercept do the exact

opposite instead of down one and right

one i go up one and left one and notice

oh there's the point of intersection

right there

i'll connect all the points with a

straight line making sure there's arrows

on both sides

i will label this one as line two

and i'm going to circle this point of

intersection right here

so our point of intersection

that point is the point negative one

three so i will say the point of

intersection and we can short form point

of intersection right that is poi the

point of intersection

is the point negative 1 3.

another way of writing your answer

instead of writing it as an x y point

you could say the solution is x equals

negative one comma y equals three those

are two different ways of communicating

the same solution now if you look at

this solution it's very well organized

and it's clear where the final answer is

when teachers are grading your work

they're required to grade your

communication of your solution as well

so make sure you're clearly

communicating the knowledge that you

have about how to solve the problem

one more thing that we should do with

each of these questions is we should

verify that the answer is correct

you can know if you got the correct

answer for these questions by checking

to make sure the x value of negative 1

and the y value of 3 satisfy both

equations let me show you how we can do

that check

to check the solution i need to check it

in both lines and i need to verify the

point is on both lines not just one

right verify it's on both lines prove

that that's where the lines intersect

making it the solution to the system so

let me check if the point negative one

three is on line 1. and how we do that

is i split the equation for line 1 into

its left side and right side

line 1 the left side is y the right side

is x plus 4.

so left side is y right side is x plus

4. if the point negative 1 3 is on this

line it'll make the left side equal to

the right side so let me just add up

here what solution i'm actually checking

i'm checking the solution x equals

negative 1 y equals 3.

so anytime i see a y i'm going to

replace it with three anytime i see an x

i'm going to replace it with negative

one and then simplify negative one plus

four is three i can see left side and

right side are the same value therefore

left side equals right side that point

is on that line

let's verify it's also on line

two

let me separate

the equation for line two into left side

right side left side is y

right side is negative x plus two

let's verify the solution change all the

y's to three change the x to negative

one so negative

negative one plus two

negative negative one is positive one

plus two is three hey that left side and

right side are equal again that means

the point is also on that line if the

point is on both of the lines that must

be where the lines intersect making it

the solution to the linear system let's

try another example we can do the next

view a little bit quicker so let's set

up our space for our work for

line

one

line one is two x plus y equals five

i want to rearrange this into the form y

equals mx plus b so i'm going to move

the 2x term to the other side of the

equation making it negative 2x and i'll

leave the positive 5 there

now you can see that the slope of this

line is equal to the m value which is

negative 2

or as a fraction we could think of it as

negative 2 over 1

and the y intercept

is equal to the b value which is

5. right here's the m

here's the b

and remember we write m as a fraction

because slope is rise

over run with that information i can

graph this line always start by plotting

the y intercept so plot five and my

slope is negative two over one rising

negative two means down two running one

means right one

down two right one and keep plotting

points until you fill the grid and to

plot points on the other side do the

opposite instead of down to right one go

up two left one

i'll connect it with a straight line and

label this line one

i will now go through the same process

for line two line two as x minus two y

equals ten

this needs to be rearranged into slope y

intercept form i'm actually going to

isolate y on the right this time

so i'm going to take this term negative

2 i move it to the right making it

positive 2y

and bring the constant 10 to the other

side making it negative 10. so i have x

minus 10 equals 2y

i want to isolate y currently it's being

multiplied by 2 so i need to divide both

sides of this equation by 2.

that means both terms on the left are be

going to be divided by 2. so i'll have a

half x

minus

10 over 2 is 5

equals 2y over 2 is y

so this is in the form mx plus b equals

y which is perfectly fine our m is a

half and our b is negative five

so that means my slope

is equal to my m value which is one over

two

and my y-intercept

is equal to my b value which is negative

five and remember slope is rise over run

so let's graph line two by first

plotting its y-intercept at negative

five

and then using the slope to plot more

points to fill the grid rise one means

up one run two means right two

and keep plotting points until we fill

the grid

and i can already see where the point of

intersection is going to be but let me

continue plotting points just to finish

off my line accurately

these two lines intersect at this point

right here that's the point four

negative three so we could write our

answer as a point of intersection

or we can write the value of x and value

of y separately

and like i said we should verify this is

the correct answer by making sure that

point 4 negative 3 is on both lines

so let me check the solution let me

verify it's on line 1 first so the

equation of line 1 was 2x plus y equals

5. so the left side is 2x plus y

and the right side is 5. and let me plug

in the x and y values that i got from my

solution x is 4

and y is negative 3.

so i have 8 plus negative 3 that means 8

minus 3 which is 5. left side equals

right side good that means the points on

that line

let's verify the point is also on the

other line and then that proves that's

the point of intersection

the equation of the other line was x

minus 2y equals 10. so the left side is

x minus 2y the right side

is 10.

let me plug in my point

4 for x negative three for y i have four

minus negative six that's four plus six

which is ten good left side equals right

side again which means the point four

negative three is on line two as well so

that's where the lines intersect okay so

the first two examples worked out very

nicely notice while we were graphing we

ended up plotting points right on top of

each other so it was very clear where

the point of intersection would be

when solving by graphing it doesn't

always work out that nicely let me show

you what i mean with part c line 1 is 2x

plus 5y equals negative 20. let me get

that into y equals mx plus b form i'll

start by moving

the 2x to the other side becomes

negative x

and then divide both sides by five

making sure to divide all terms by five

so negative two x over five is negative

two over five

times x

minus twenty divided by five is four

so there's the equation in y equals mx

plus b form where m is negative two over

five

and b is negative four so my slope is

negative two over five and my

y-intercept is negative four so to graph

this line we always start by plotting

the y intercept

and then use the slope rise negative two

run five to plot more points rise

negative two means down two run five

means right five

i won't be able to plot very many points

on this graph but i'll plot as many as i

can

to the other side of the y intercept do

the opposite instead of down two right

five go up two

and left five

and i've connected the points and

labeled the line so there's my line one

let's graph line two and see where they

intersect

line two is five x minus three y equals

negative fifteen

i'm going to rearrange this one to

isolate y on the right this time so i'll

have 5x

plus 15 equals 3y

and divide both sides by 3 i get 5 over

3x plus 15 over 3 which is 5 equals 3y

over 3 which is y

so there we have it slope y intercept

form

where m is 5 over 3

b is 5.

so my slope

is 5 over 3 and my y intercept is 5. let

me graph this line by starting by

plotting the y intercept and using the

slope of rise five

run three

so up five right three and the opposite

down five left three

there's my line 2 graphed

and now i'm going to try and find the

point of intersection

now the lines definitely cross

but my answer for this is going to have

to be an estimation they cross at this

point right here

what do i think the coordinates of that

point are

i'm not sure exactly but if i were to

estimate i'd say it looks like it's at

somewhere between negative 4 and

negative 5 as the x value

and somewhere between negative 2 and

negative 3 is the y value so i'm just

going to estimate

the point of intersection i think is

approximately

negative 4.3

comma negative 2.2

about

or another way of writing this the

solution

is once again i'll write the word

approximately to show that this is an

estimation

x equals negative 4.3

y equals negative 2.2

now i'm not going to try and verify this

solution because this has been an

estimation

this point negative 4.3 negative 2.2

might not be on either of the lines but

if i were to sub them back in and check

it should make left side and right side

of both equations like

approximately equal to each other but

there's no way to verify for this type

of solution if it's exactly correct or

not which is why in lesson two and three

we need to learn a better way of solving

linear systems so that we don't have to

do any estimation we'll learn algebraic

methods of substitution and elimination

that avoid us having to do any

estimation now remember when solving

linear systems you don't always get one

solution like we got in a b and c

part d and e are going to show you the

other scenarios

where you could get no solutions or

infinite solutions let's see which one

part d is so let's graph both lines i

mean i can tell right away that these

have the same slope but different

y-intercepts so i know they're parallel

and distinct and are going to have no

solutions but let's verify that

graphically

so line one is already in the form y

equals mx plus b

where our m value is our slope

and it's two which is two over one

and our y intercept is the b value which

is

3. so if i had to graph this line i

would plot the y-intercept of 3 and use

my slope of 2 over 1 to plot more points

now let's look at the properties of line

2.

once again line 2 is already in the

format y equals mx plus b

where our slope is equal to the m value

of

two which we could think of once again

as two over one

and our y intercept

equals the b value of the equation which

is negative four so if i were to graph

this one by plotting the y-intercept and

using the slope

notice i get a line that runs exactly

parallel to line one so it's never going

to intersect it therefore there are no

solutions to this linear system

so our final answer for this would be

the lines are parallel and distinct

therefore there are no solutions

or you could say there is no point of

intersection

last example let's work with line one

which is x plus y equals three let me

rearrange this into y equals mx plus b

form by moving that positive x to the

right becomes a negative x

now i can see that my slope

is equal to the m value the coefficient

of that x is negative one which as a

fraction is negative one over one

the y intercept

is equal to the b value which is

three

to graph this line plot the y intercept

and use the slope of rise negative one

run one that means down one right one

and then line two i have two x plus two

y equals six if i rearrange this one to

isolate y

i would have negative two x plus six on

the right

and then isolate the y by dividing both

sides by two so divide all terms by two

i'd get y equals negative x plus three

hey

notice exact same equation as line one

that tells me it's going to have the

same slope and the same y-intercept so

when i graph this one plot the

y-intercept

and then use the slope to plot more

points notice all the points are right

on top of each other all of these points

are solutions to the linear system so

from the equations i can see they have

the same slope and y intercept

meaning they're going to be parallel and

coincident meaning there are infinitely

many solutions to this system so let me

communicate that in my final answer oh

and i suppose i should somehow try and

show line two right on top of line one

so maybe

i'll put it right on top

and see if i can make it a little bit

transparent so we can see through to the

other one and i'll label that as line

one and line two so let me now write my

final answer the lines are parallel and

coincident

therefore there are infinitely many

solutions and that's it for this lesson

stay tuned for lesson two where we learn

the method of substitution to solve

linear systems