Solve Linear Systems by GraphingSolve Linear Systems by SubstitutionSolve Linear Systems by Elimination

Adding or Subtracting Multiplying First

Solve Special Type of Linear SystemsSolve systems of Linear Inequalities

Systems of Linear Equation

Solve Linear Systems

by Graphing

The cost to join an art museum is P600. If you are a member, you can take a lesson at the museum for P20 each. If you are not a member, lessons cost P60 each. Write an equation to find the number of x of lessons after which the total cost y of the lessons with membership is the same as the total cost of lessons without a membership.

Getting Ready!

1. Complete the table below for x + y = 4.

X 0 1 2 3 4 5

y

2. Complete the table below for 2x – y = 5.

X 0 1 2 3 4 5

y

Can you name a pair that satisfies

both equation?

4 3 2 1 0 -1

-5 -3 -1 1 3 5

System of Linear EquationsLinear systemConsist of two or more linear

equations in the same variablesx + 2y = 7 and 3x – 2y = 5

Solution of a System of Linear Equations

Is an ordered pair that satisfies each of the equation in the systems

x + y = 42x – y = 5

(3, 1)

X 0 1 2 3 4 5

y 4 3 2 1 0 -1

X 0 1 2 3 4 5

y -5 -3 -1 1 3 5

Point of intersection

(3,1)

1) -5x + y = 0 and 5x + y = 10y = -5x +10

(1, 5)

Point of

intersection

3.Check1.Graph

2.Identify the point

of intersectiony = 5x5 = 5(1)5 = 5

y = 5x

y = -5x + 105 = -5(1) + 105 = -5 +105 = 5

(1,5) is a solution of the Linear system

2) x – y = 5 and 3x + y = 3y = -3x + 3

(2, -3)

Point of

intersection

3.Check1.Graph

2.Identify the point

of intersection y = x – 5-3 = 2 – 5-3 = -3

y = x - 5

y = -3x + 3-3 = -3(2) + 3-3 = -6 + 3-3 = -3

(2, -3) is a solution of the Linear system

Consistent Independent system of Linear equation

It has at least one solution.

It has

different

graphs.

F.Y.I.:

Inconsistent system of linear equations

– does not have a solution.

Dependent system of linear equations

– equations with identical graph.

Steps in Solving for Systems by Graphing

1. Graph both equations in the same coordinate plane.

2. Identify the point of intersection.

3. Check the coordinates algebraically by substituting it in to each equation.

3) x + y = 4 and 2x – y = 5y = 2x - 5

(3, 1)

Point of

intersection

3.Check1.Graph

2.Identify the point

of intersectiony = -x + 41 = -(3) + 41 = 1

y = -x + 4

y = 2x - 51 = 2(3) – 51 = 6 - 51 = 1

(3,1) is a solution of the Linear system

4) x – y = 1 and x + y = 3y = -x + 3

(2, 1)

Point of

intersection

3.Check1.Graph

2.Identify the point

of intersection y = x – 11 = 2 – 11 = 1

y = x - 1

y = -x + 31 = -(2) + 31 = -2 + 31 = 1

(2, 1) is a solution of the Linear system

HomeworkSolve the following linear systems by graphing.

1. y = -x + 3 and y = x + 12. 3x + y = 15 and y = -15

1. y = -x + 3 y = x + 1

2. 3x + y = 15 y = -15

Solve Linear Systems by Substitu

tion

A gardening company placed orders with a nursery. One was for 13 bushes and 4trees, and totaled P487. The second order was for 6 bushes and 2 trees, and totaled P232. The bill doesn't tell the amount of per item. What were the costs of one bush and of one tree?

Getting Ready!

Simplify the following:1. 3(2x + 3)2. -4(3x – 4)

Can you solve the system of

linear equation by

substitution?

Substitute x – 3 for y and simplify the following:

1. 5y2. 3y + 23. 2(y+3)Solve for x and y 1. y = 2x + 1 and 3x + 2y = 9

y=2x + 1 and 3x + 2y = 9 1. Solve for

a variable

Eq. 1 Eq. 2

Eq. 1 is already

solved for y.

2. Substitute Eq.1 to Eq. 23x + 2y = 9 eq. 23x + 2(2x + 1) = 9

3x + 4x + 2 = 97x = 9-27x = 7

x = 1

3. Substitute value of x to eq. 1y = 2x + 1 eq. 1y = 2(1) + 1y = 2 + 1y = 3

The solution is (1, 3).

a + b = 7 and 3a + 2b = 161. Solve for a variable

Eq. 1 Eq. 2

2. Substitute Eq.1 to Eq. 2

3a + 2b = 16 eq. 23a + 2(-a + 7) = 16

3a – 2a + 14 = 16a = 16 - 14a = 2

3. Substitute value of a to eq. 1b = -a + 7 eq. 1b = -(2) + 7b = -2 + 7b = 5

The solution is (2, 5).

a + b = 7 eq. 1b = -a + 7

Steps in Solving for Systems by Substitution

1. Solve for one variable using one of the equations.

2. Substitute the expression from step 1 into other equation and solve.

3. Substitute the value from step 2 into the expression of step 1 and

solve.

x – 2y = -6 and 4x + 6y = 41. Solve for

a variable

Eq. 1 Eq. 2

2. Substitute Eq.1 to Eq. 24x + 6y = 4 eq. 24(2y – 6) + 6y = 4

8y – 24 + 6y = 414y = 4 + 2414y = 28

y = 2

3. Substitute value of x to eq. 1x = 2y - 6 eq. 1x = 2(2) - 6x = 4 - 6x = -2

The solution is (-2, 2).

x – 2y = -6x = 2y - 6

m = 2n + 5 and 3n + m = 101. Solve for a variable

Eq. 1 Eq. 2

2. Substitute Eq.1 to Eq. 2

3n + m = 10 eq. 23n + (2n + 5) = 10

3n + 2n + 5 = 105n = 10 - 5

n = 13. Substitute value of x to eq. 1

m = 2n + 5 eq. 1m = 2(1) + 5m = 2 + 5m = 7 The solution is (1, 7).

Eq. 1 is already solved

for m.

5n = 5

HomeworkSolve the following linear systems by substitution

1. x = y + 3 and 2x – y = 5

2. 11a – 7b = -14 and a- 2b =-4

SeatworkSolve each of the following linear

systems by substitution and graph.1. 2x – y = 4 and 3x + 2y = -42. 3x = 2y + 5 and y + 1 = 03. x + 2y = 3 and x – y = 64. 3x = 2y + 8 and y = 3x

Solve Linear Systems by Eliminati

on

Solve Linear System by ADDING or

SUBTRACTING

Getting ready!Solve for the following:

(3x -2y) + (3x + 3y)=

(3x - 4y) + (-3x + 2y)=

(x + y) - (x - 4y)

(-2x - 4y) + (-4x + 4y)

Developing SkillsAdd the following equation.

Which variable is

eliminated?

x + 2y = 53x – 2y = -1

4x = 4x = 1

y

Solve for the remaining variable.

x + 2y = 51 + 2y = 5

2y = 5-1 2y = 4 y = 2

The solution is (1, 2)

Add the following equation.

Which variable is

eliminated?

2a – 3b = 14 a + 3b = -2

3a = 12a = 4b

Solve for the remaining variable.

a + 3b = -2 4 + 3b = -2

3b = -2 - 4 3b = -6

b = -2

The solution is (4, -2)

Developing SkillsAdd the following equation.

Which variable is

eliminated?

2x + 3y = 11 2x - 5y = 13

8y = 24y = 3

x

Solve for the remaining variable.

2x + 3y = 11 2x + 3(3) = 11

2x = 11 - 9 2x = 2 x = 1

The solution is (1, 3)

Steps in Solving for Systems by adding

and subtracting1. Add or subtract the equations to

eliminate one variable.

2. Solve for one variable.

3. Substitute the value from step 2 into one original equation and solve.

Add the following equation.

Which variable is

eliminated?

2a – 3b = 26 -2a - 3b = -2

-6b= 24b = -4b

Solve for the remaining variable.

2a - 3b = 26 2a – 3(-4) = 26

2a = 26 - 12 2a = 24

a = 12

The solution is (12,-4)

Solve Linear System by

MULTIPLYING FIRST

5x + 2y = 163x – 4y = 20

Can we eliminate a variable by adding and

subtracting?

(5x + 2y = 16)210x + 4y = 323x – 4y = 20

10x + 4y = 32 3x – 4y = 20

13x = 52x = 4 3x – 4y = 20

3(4) – 4y = 20 – 4y = 20 -12 – 4y = 8

y = -2

The solution

is (4, -2)

6x + 5y = 19-6x - 9y = -15

-4y = 4y = -1

6x + 5y = 19 6x +5(-1) = 19

6x = 19 + 5 6x = 24

x = 4 The solution is (4, -1)

6x + 5y = 19 2x + 3y = 5 ( )-3

8x + 10y = 70-15x+10y = -45

23x = 115x = 5

4x + 5y = 35 4(5) +5y = 35

5y = 35 - 20 5y = 15

y = 3

The solution

is (4, -1)

4x + 5y = 35 -3x + 2y = -9 ( ) 5 ( ) 2

SeatworkSolve each of the following linear

systems by Elimination.

1. ) 6x – 2y = 1 -2x + 3y = -5

2.) 2x + 5y = 3 3x + 10y = -3

HomeworkSolve the following linear systems by Elimination

1. 3x - 7y = 5 and 9y= 5x + 5

2. 3a + 2b = 4 and 2b =8 – 5a

Short QuizSolve the following linear systems by Elimination

1. x + 4y = 22 and 4x – y = 32. 2x – 3y = 10 and x + 3y = -83.3x – y = 5 and 5x + 2y = 23

Solve Special Type of Linear

Systems

Consistent Independent system of Linear equation

It has at least one solution.

It has

different

graphs.

Inconsistent Linear System> A linear system that has no solution.

3x + 2y = 103x + 2y = 2

No solution

The slopes of an

inconsistent linear system

is equal.

Dependent Linear SystemA linear system that has infinitely many solution.It has identical graph x – 2y = -4

y = (1/2)x + 2

Infinitely many solution

The slopes and y-intercept is

equal

Number of solutions

Slopes and y - intercept

One solution Different slopes

No solution Same SlopeDifferent y-intercept

Infinitely many solutions

Same SlopeSame y-intercept

Bell WorkDetermine whether the statement is true or false. 1. A solution of a linear system is an ordered pair (x,y)

2. Graphically, the solution of an independent system is the point of intersection.

3. An independent system of equation has no solution.

4. The graph of an inconsistent system is identical.

5. A system of linear equation can either have one solution or no solution.

II. Answer the following.

6. Is (2,3) a solution of the system 3x + 4y = 18

2x – y = 1 ? 7. Is ( 1, -2 ) a solution of the system 3x – y = 14 2x + 5y = 8 ?

8. Is (-1,3) a solution of the system 4x – y = -5

2x + 5y = 13 ? 9. Is (0,0) a solution of the system

4x + 3y = 0 2x – y = 1 ?

10. Is (2,-3) a solution of the system y = 2x – 7 3x – y = 9 ?

Seatwork: Without graphing, determine whether the linear system is independent, inconsistent, or dependent. 1. y = -9x + 5 2. 3x + y = 6

y = 4x – 8 3x + y = -8

3. x + y = 3 4. x – y = 3 2x + 2y = 6 x + y = 5

Seatwork: Without graphing, determine whether the linear system is independent, inconsistent, or dependent. 1. y = -9x + 5 2. 3x + y = 6

y = 4x – 8 3x + y = -8

3. x + y = 3 4. x – y = 3 2x + 2y = 6 x + y = 5

5. 3x – y = 3 6. 2x – y = 42x + y = 2 x + y = 5

7. x + 2y = 6 8. 5x – 2y = 10 x – 2y = 3 3x + 2y = 6

9. 4x – 5y = 20 10. 8x + 2y = 6 8x – 10y = 12 y + 4x = -1

SeatworkSolve each of the following

linear systems by any method and identify the type of system.

1. ) y = -6x - 2 12x + 2y = -6

2.) 9x – 15y = 24 6x – 10y = 16

HomeworkSolve the following linear systems and indentify the type of system.

1. y = 7x + 13 and -21x + 3y = 39

2. x – 2y = 7 and –x + 2y = 7

Solve Systems of Linear

Inequalities

System of Linear Inequalities• Consist of two or more linear inequalities in

the same variable.

Solution of a System of Linear Inequalities

• Is an ordered pair that is a solution for both linear system.

Graph of a System of Linear Inequalities

• Is the graph of all solutions of a system.

y > -x – 2 and y ≤ 3x + 6

y > -x – 2 and y ≤ 3x + 6

x > 1 and x <4

x > 1 and x <4

y > -3 and y < 4

y > -3 and y < 4

Steps in Graphing a system of Linear Inequalities

Step 1: Graph each inequality.Step 2: Find the intersection of the shaded part.

The graph of a system is its intersection.

y ≥ -1, x > -2 and x + 2y ≤ 4

y ≥ -1, x > -2 and x + 2y ≤ 4

SeatworkSolve each of the following linear

Inequalities1. ) y ≤ 5x + 1

y > x - 2

2.) y > -1 y < 4

y ≤ 5x + 1 and y > x - 2

y > -1 and y < 4

HomeworkSolve the following linear inequalities.

1. y ≤ x – 3 and y > -2x - 12. x < 8 and x > -2

y ≤ x – 3 and y > -2x - 1

X < 2 and x > -2

Problem Solving1. There are 20 animals in a pen compose of dog and duck. If the number of their legs is 52, how many dogs are there? How many chickens are there? Let x be the number of dogs y be the number of ducksWORKING EQUATION:

X + Y = 204X + 2Y = 52

X + Y = 204X + 2Y = 52( ) 2

2X + 2Y = 404X + 2Y = 52

-

-2X = -12X = 6

X + Y = 206 + Y = 20

Y = 20 - 6Y = 14

Therefore, there are 6 dogs and 14 chickens.

2. A gardening company placed orders with nursery. One was fo

Let x be the number of dogs y be the number of ducksWORKING EQUATION:

X + Y = 204X + 2Y = 52

X + Y = 204X + 2Y = 52( ) 2

2X + 2Y = 404X + 2Y = 52

-

-2X = -12X = 6

X + Y = 206 + Y = 20

Y = 20 - 6Y = 14

Therefore, there are 6 dogs and 14 chickens.