Intro to Systems of Equations and Graphing

Intro to Systems of

Equations and Graphing

byTammy Wallace

Varina High School

What is a System of Equation?

A SYSTEM OF EQUATIONS is:

A SOLUTION to the SYSTEM OF EQUATIONS is:

A POINT OF INTERSECTION is:

two or more equations with the same variable being compared.

the point that both equations have in common: THE POINT OF INTERSECTION

the point where both lines meet

Write the solution of each system of equation as an

ordered pair.

(4, -3)

(8, -3)

Both systems have only ONE SOLUTION.


ordered pair.

(2, -2) (2,

1)


ordered pair.

∅ ∅

Parallel lines do not intersect. Therefore, there is NO SOLUTION!


ordered pair.What is different about this graph? There appears to only be one line graphed. We call this coinciding lines.

Because there are more than one line on top of each other, what can we conclude about the solution set?There are infinite many solutions

ℝ


ordered pair.

(-3, 0)

Graphing System of Equations

When finding the solution to a system of equation, finding the point of intersection is required.

To do this, the equations must be graphed. Therefore, the equation must be put in SLOPE-INTERCEPT FORM.

𝑦=𝑚𝑥+𝑏

Procedures

Put each equation in Slope-Intercept Form

Identify the slope and y- intercept

m = ____ b = ____

m = ____ b =____

Find the solution set for {𝒚=−𝟏𝟐 𝒙+𝟒

𝟑 𝒙−𝟐 𝒚=𝟖

Already in the correct format!

−𝟏𝟐 4

𝟑𝟐 -4

3x – 2y = 8 -3x - 3x

-2y = -3x + 8 -2 -2

Draw each line on the same grid.

Procedures


m = ____ b = ____

m = ____ b =____−𝟏

𝟐 4𝟑𝟐 -4

a) What type of lines are there?

b) What type of solutions do these lines have?

c) What is/are the solution(s) located?

Intersecting lines

One solution

( 4, 2 )

Procedures



m = ____ b = ____

m = ____ b =____

Find the solution set for {𝟐 𝒙+𝒚=𝟒𝒙− 𝒚=𝟐

2x + y = 4 - 2x -2x

y = -2x + 4

−𝟐 4 𝟏 -2

x - y = 2 -x -x

-y = -x + 2

y = x - 2

-1( )

Draw each line on the same grid.

Procedures


m = ____ b = ____

m = ____ b =____−𝟐 4 1 -2


b) What type of solutions do these lines have?

c) What is/are the solution(s) located?

Intersecting lines

One solution

( 2, 0)

x − y=2

ProceduresPut each equation in Slope-Intercept Form


m = ____ b = ____

m = ____ b =____

Find the solution set for { 𝒚=𝟐 𝒙−𝟑𝟐 𝒙−𝒚=−𝟏

Already in the correct format!

𝟐 -3 𝟐 1

2x – y = -1 -2x - 2x

-y = -2x - 1

-1( )

Graph each equation in the graphing

calculator

and answer the following questions. Procedures


m = ____ b = ____

m = ____ b =____

𝟐 -3 𝟐 1


b) What type of solutions are there?

c) What do you notice about the equations when they are in slope intercept form AND do you think this has an effect on the solution?

Parallel lines

NO SOLUTIONS

When the equations are in slope-intercept form, you notice the slopes are the same. When the slopes are the same, the lines will be parallel and have solution.

Procedures 3x



m = ____ b = ____

m = ____ b =____

Find the solution set for { 𝟑 𝒙−𝒚=𝟖𝟐 𝒚=𝟔 𝒙−𝟏𝟔

3x - y = 8 - 3x -3x

-y = -3x + 8

y = 3x - 8𝟑 -8 𝟑 -8

2y = 6x - 16 2 2

y = 3x - 8-1( )

Graph each equation in the graphing

calculator

and answer the following questions. Procedures


m = ____ b = ____

m = ____ b =____

𝟑 8 𝟑 8


b) What type of solutions are there?

c) What do you notice about the equations when they are in slope intercept form AND do you think this has an effect on the solution?

Coinciding lines

Infinite many solutions

When the equations are in slope-intercept form, you notice the equations are the same. When the equations are the same, there are infinite many solutions.

Application of Systems of Equations

When you are given a word problem, remember to do the

following:

Equation in Words

Variables

Equation with Numbers/Variab

les

Equation in Slope-

Intercept Form

1)

2)

Rewrite what the first equation could be.

Rewrite what the second equation could be.

Determine the

what the unknown variables are and

what letter

they will represent

.

Rewrite the first equation algebraically

using the variables chosen.

Rewrite the second

equation algebraically

using the variables chosen.

Rewrite the second

equation in slope-intercept

form and graph.

Rewrite the first equation

in slope-intercept form

and graph.

Finally, find the solution set using the calculator.

The school is selling tickets to a band concert. On the first day of ticket sales, the school sold 3 adult tickets

and 1 student tickets for a total of $28. The school took in $32 on the second day by selling 3 adult tickets and 2 student tickets. Find the price of a

student ticket and an adult ticket.

Equation in Words

Variables


les

Equation in Slope-

Intercept Form

1)

2)

The school sold 3 adult tickets and 1 student ticket for $28

The school sold 3 adult tickets and 2 student ticks for $32

a = cost of adult tickets(let a = x)

s = cost of

student tickets

(let s = y)

3x + y = 28

3x + 2y = 32

3x + y = 28

- 3x -3x y = -3x + 283x + 2y =

32- 3x -

3x 2y = -3x + 32 2 2

The school is selling tickets to a band concert. On the first day of ticket sales, the school sold 3 adult tickets

and 1 student tickets for a total of $28. The school took in $32 on the second day by selling 3 adult tickets and 2 student tickets. Find the price of a

student ticket and an adult ticket.

Equations in Slope-Intercept Form

Graph of the System 1)

y = -3x + 28

2)

Solution set is (8,

4)

Graph each equation. Where is the solution set located?

Remember what x and y equals when it relates the word problem:

x = ______________ and y = _____________

What does the solution set mean?

The solution is located at (8, 4).

cost of adult tickets

cost of student tickets

Adult tickets cost $8 each and student tickets cost $4 each.

You are getting ready to move and have asked some friends to help. For lunch, you buy the following sandwiches at the local deli for $30: six ham sandwiches and six turkey sandwiches.

Later at night, everyone is hungry again and you buy four ham sandwiches and eight turkey sandwiches for $30.60. What is

the price of each sandwich?

Equation in Words

Variables


les

Equation in Slope-

Intercept Form

1)

2)

6 ham and 6 turkey for $30

4 ham and 8 turkey for

$30.60

h = ham sandwiche

s(let h = x)

t = tuna sandwiche

s(let t = y)

6x + 6y = 30

4x + 8y = 30.20

6x + 6y = 30- 6x -6x 6y = -6x + 30 6 6 y = -x + 5

4x + 8y = 30.20

-4x -4x 8y = -4x + 30.20 8 8

Equations in Slope-

Intercept FormGraph of the System

1)

y = -x + 5

2)

You are getting ready to move and have asked some friends to help. For lunch, you buy the following sandwiches at the local deli for $30: six ham sandwiches and six turkey sandwiches.

Later at night, everyone is hungry again and you buy four ham sandwiches and eight turkey sandwiches for $30.60. What is

the price of each sandwich?

Solution set is (2,

3)

Graph each equation. Where is the solution set located?

Remember what x and y equals when it relates the word problem:

x = ______________ and y = _____________

What does the solution set mean?

The solution is located at (2, 3).

cost of ham sandwiches

cost of turkey sandwiches

Ham sandwiches cost $2 each and turkey sandwiches cost $3 each.

The concession stand is selling hot dogs and hamburgers during a game. At halftime, they sold a total of 50 hot dogs and

hamburgers and brought in $105.50. How many of each item did they sell if hamburgers sold for $1.50 and hot dogs sold for

$1.25?

Equation in Words

Variables


les

Equation in Slope-

Intercept Form

1)

2)

50 hotdog and hamburgers were sold

With selling hotdogs for .50

and hamburgers for

$1, $105.50 was made.

h = hotdogs(let h = x)

b = hamburger

s(let b = y)

x + y = 50

0.5x + y = 75.50

x + y = 50 - x -x y = -x + 50

0.5x + y = 75.5 -0.5x -0.5x y = -0.5x + 75.5

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Intro to Systems of Equations and Graphing

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