Chapter 5 Systems of

Equations and Inequalities

Ms. Fisher

Monday 11-23-2014 Chapter 5 page 329Objective: Identify Solutions of systems of linear equations in two variables.Lesson 5.1Example: x+2y= 6 Is (4,1) a solution of the given system? x- y= 3 Think about what we already know…Solution means… A value of a variable that makes the equation true Linear equation means…. An equation whose graph is a straight line

A System of Linear Equations is s set of two or more linear equations that each contain two or more variables. In this course, the systems consist of two equations that each contain two variables. A solution of such a system is an ordered pair that satisfies both equations. Consider the system: x+y= 5 The ordered pair (3,2) is a soln of y-x= -1 the system bc it satisfies BOTH equations! 3+2=5 2-3= -1

If an ordered pair is a solution it must lie on both graphs. Therefore, the solution is the intersection of the two graphs.

x+y=5

y-x=-1

x+2y= 6 Is (4,1) a solution of the given system?x- y= 3

x + 2y = 6 x-y= 34 + 2(1) =6 4-1=34+2=6 Yes 3=3 Yes

The ordered pair (4,1) makes both equations true.Therefore, (4,1) is a solution of the system.

Solve & Graphy= 2x – 1 Is point (2,3) a solution of the given system?y= -x + 5

y=2x-1 y= -x +53=2(2) -1 3= -2 +53=4-1 Yes 3=3 YesTo Graph:Write in slope intercept form: y= mx + bb= your y interceptFor line y=2x-1 it intercepts at (0,-1) & (2,3)For line y= -x + 5 it intercepts at (0,5) & (2,3)Now, simply draw both lines!

y=2x-1

y=-x+5

Independent Work Time

Small group with Ms. Fisher: Maddie, Nikki, & Christian HW: pg 332 #’s 1-7Extension: Alex & James HW & #’s 9-15Leo: Study Island on laptop

Tuesday 11-24-2014 Chapter 5 page 336Agenda:1. Go over homework - Any Questions please park in the Parking Lot on front board!2. Teach lesson 5.2 Whole Group3. Independent Work Time4. Small Group Focus Time with Ms. Fisher

Objective: Solve systems of linear equations in two variables by substitutions.New Concept: SubstitutionWhat do we think that means??????? Prior knowledge???

Last nights HW: pg 332 #’s 2-7

Any Questions please park in the parking Lot on front board!

Tell whether the ordered pair is a solution of the given system?2. No3. Yes4.YesSolve each system by graphing5. (2,1)6.(1,-1)7.(-4,7)

5.2 Solving Systems by Substitutions pg 336

The goal when using substitution is to reduce the system to one equation that has only one variable. Then you can solve that equation. Then you substitute the answer you found in that equation back into the original equation to find the answer to the other variable!Step #1: Reduce the system, Solve for one variable (ex: y) y=2xStep #2: Substitute the expression/ “solved variable”(y) into the other equation.Step #3: Solve the equation to get the value of the first variable (x)Step #4: Substitute the value of the first variable into one of the original equations to find the value of the second variable.Step #5: Write the values as an ordered pair, (x,y)

Example: Solve each system by substitutiony=2xy= x+5Step #1: Reduce the system, Solve for one variable y=2xStep #2: Substitute the expression into the other equation2x= x +5Step #3: Solve the equation to get the value of the first variable (x)2x= x +5-x -xx=5Step #4: Substitute the value of the first variable into one of the original equations to find the value of the second variable.y=2x Step #5: Write values as ordered pairy=2(5) (5,10)y=10

Turn to page 340

As a whole group try #1Solve each system by substitution:1. y=5x-10 Answer: (9,35) y=3x+8Independent Work:Small Group: Maddie, Nikki, Christian W/ Ms. Fisher HW #’s 2-6Extension: Alex & James HW & #’s 8-16

December 1 Monday Agenda

Park HW Questions in parking Lot on front board1. Go over HW pg 340 #’s 2-63. Teach lesson 5.3 Whole Group Lesson4. Small Group Focus5. Independent Work

Objective: Solve systems of linear equations in two variables by elimination.New Concept: Elimination Call on prior English knowledge…What do you think elimination might mean when working in Math?

HW page 340 #’s 2-6

2. (18,-52)3. (3,8)4. (12,1)5. (-3,-9)6. (-4,4)

5.3 Solving Systems by Elimination

-The goal of elimination is to get one equation that has only one variable.-When you use the elimination method to solve a system of linear equations, align all like terms in the equations. Then determine whether any like terms can be eliminated because they have opposite coefficients.-If you don’t have a like term to eliminate, create like terms! Remember an equation stays balanced if you do it to both sides.

5x +2y = 1 x + 3y = 1 What would you do? -2x- 6y= -2 x -2y = -19 2x +2y = 1 (x+3y =1) -2 2x +2y= 1 6x + 0 = -18 -2x - 6y =-2 -4y= -1

Solve each system by elimination: Answer (3,-3)2x + y = 3-x +3y = -12Step #1: Multiple each term in the second equation by 2 to get the opposite x-coefficient.2(-x +3y= -12) -2x +6y =-24Step #2: Add the new equation to the first equation to eliminate x2x + y= 3 Step #3: Substitute -3 for y in one of the -2x+6y= -24 original equations to find x 2x +y =3 7y= -21 y= -3 2x -3 =3 2x=6 x=3

Independent Practice:Small Group with Ms. Fisher Maddie, Nikki, & Christian page 347 #’s 1-9Extension: Alex & James HW & 11-19Leo: Study Island

December 2 Tuesday Agenda

Park HW Questions in parking Lot on front board1. Go over HW pg 347 #’s 1-9 2. Briefly Discuss 5.4 Whole Group Lesson- OMIT4. Introduce Quiz #1 SLO- Student Learning Objective

New State Requirement5. If times allows Quiz #2 SLO

Objective: A1.1.2.2 Write, solve, and/or graph systems of linear equations using various methods.

HW pg 347 #’s 1-9 1. (-4, 1)2. (7,5)3. (-2,-4)4. (40,-2)5. (-6,30)6. (-5,-2)7. (3,2)8. (-4,0)9. (4,-3)

December 3rd Wednesday Agenda

1. Teach Lesson 5.5 Whole Group2. Small Group Focus Work3. Independent Work

Objective: Graph and Solve linear inequalities in two variables

Prior knowledge… What do you know about the wordInequality????? A statement that two quantities are not equal≤ ≥ < >

5.5 Solving Linear Inequalities

A linear inequality is similar to a linear equation but the equal sign is replaced with an inequality symbol.Example:Is the following ordered pair a solution of the inequality?y< x-1 (7,3)3< 7-13< 6 Yes, true statement so is a solution

y intercept

Plug 0 in for yY=3x + 40< 3x +4 x= -1 1/3

X intercept

Independent Practice:Small Group with Ms. Fisher Maddie, Nikki, & Christian page 364 #’s 2,3,4, 5-8, 10 &11Extension: Alex & James: HW & 15-18, 20 & 21

December 4th Thursday Agenda LAST Section in Chapter 5!

Park HW Questions in parking Lot on front board1. Go over page 364 #’s 2,3,4, 5-8, 10 &112. Teach Lesson 5.6 Whole Group3. Small Focus Group4. Independent Work

Objective: Graph and solve systems of linear inequalities in two variablesWhat do we already know about linear inequalities???!!!

HW 364 #’s 2,3,4, 5-8, 10 &11

2. Yes 6. y= 3x +13. Yes4. No5. y≤ -x + 0

6. 10. y < 37. 11. y ≥ x+58.

5.6 Solving Systems of Linear Inequalities

Is the ordered pair a solution of the given system?

y < -x + 4 (2,1)y ≤ x+1

y< -x + 4 y ≤ x + 11< -2 +4 1 ≤ 2 + 11 < 2 Yes 1 ≤ 3 Yes Satisfies BOTH inequalities So point (2,1) is a solution!

Independent PracticePage 370 #’s 2-14 HW Small group: Maddie, Christian, Nikki work with Ms. FisherExtension: Alex & James HW & 23-28

Review for Chapter 5 Test !!

Page 375 #’s 1-20