Chapter 5
Systems of Equations
and Inequalities
Ms. Fisher
Monday 11-23-2014 Chapter 5 page 329
Objective: Identify Solutions of systems of linear
equations in two variables.
Lesson 5.1
Example: 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
y-x=-1
equations! 3+2=5
2-3= -1
x+y=5
If an ordered pair is a solution it must lie on both graphs.
Therefore, the solution is the intersection of the two graphs.
x+2y= 6
x- y= 3
Is (4,1) a solution of the given system?
x + 2y = 6
4 + 2(1) =6
4+2=6 Yes
x-y= 3
4-1=3
3=3 Yes
The ordered pair (4,1) makes both equations true.
Therefore, (4,1) is a solution of the system.
Solve & Graph
y= 2x – 1 Is point (2,3) a solution of the given system?
y= -x + 5
y=2x-1
y= -x +5
3=2(2) -1
3= -2 +5
3=4-1 Yes
3=3 Yes
To Graph:
Write in slope intercept form: y= mx + b
b= your y intercept
For 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 1-7
Extension: Alex & James HW & #’s 9-15
Leo: Study Island on laptop
Tuesday 11-24-2014
Chapter 5 page 336
Agenda:
1. Go over homework Any Questions please park in the Parking Lot on front board!
2. Teach lesson 5.2 Whole Group
3. Independent Work Time
4. Small Group Focus Time with Ms. Fisher
Objective: Solve systems of linear equations in two variables by
substitutions.
New Concept: Substitution
What 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. No
3. Yes
4.Yes
Solve each system by graphing
5. (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=2x
Step #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 substitution
y=2x
y= x+5
Step #1: Reduce the system, Solve for one variable
y=2x
Step #2: Substitute the expression into the other equation
2x= x +5
Step #3: Solve the equation to get the value of the first variable (x)
2x= x +5
-x -x
x=5
Step #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 pair
y=2(5)
(5,10)
y=10
Turn to page 340
As a whole group try #1
Solve each system by substitution:
1. y=5x-10 Answer: (9,35)
y=3x+8
Independent Work:
Small Group: Maddie, Nikki, Christian W/ Ms. Fisher HW #’s 2-6
Extension: Alex & James HW & #’s 8-16
December 1 Monday Agenda
Park HW Questions in parking Lot on front board
1. Go over HW pg 340 #’s 2-6
3. Teach lesson 5.3 Whole Group Lesson
4. Small Group Focus
5. 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 -2y = -19
6x + 0 = -18
x + 3y = 1 What would you do? -2x- 6y= -2
2x +2y = 1 (x+3y =1) -2
2x +2y= 1
-2x - 6y =-2
-4y= -1
Solve each system by elimination: Answer (3,-3)
2x + y = 3
-x +3y = -12
Step #1: Multiple each term in the second equation by 2 to get the
opposite x-coefficient.
2(-x +3y= -12)
-2x +6y =-24
Step #2: Add the new equation to the first equation to eliminate x
2x + 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 1-9
Extension: Alex & James HW & 11-19
Leo: Study Island
December 2 Tuesday Agenda
Park HW Questions in parking Lot on front board
1. Go over HW pg 347 #’s 1-9
2. Briefly Discuss 5.4 Whole Group Lesson- OMIT
4. Introduce Quiz #1 SLO- Student Learning Objective
New State Requirement
5. 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 Group
2. Small Group Focus Work
3. Independent Work
Objective: Graph and Solve linear inequalities in two variables
Prior knowledge… What do you know about the word
Inequality????? 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< 7-1
3< 6 Yes, true statement so is a solution
y intercept
X intercept
Plug 0 in for y
Y=3x + 4
0< 3x +4 x= -1 1/3
Independent Practice:
Small Group with Ms. Fisher Maddie, Nikki, &
Christian page 364 #’s 2,3,4, 5-8, 10 &11
Extension: Alex & James: HW & 15-18, 20 & 21
December 4th Thursday Agenda LAST Section in Chapter 5!
Park HW Questions in parking Lot on front board
1. Go over page 364 #’s 2,3,4, 5-8, 10 &11
2. Teach Lesson 5.6 Whole Group
3. Small Focus Group
4. Independent Work
Objective:
Graph and solve systems of linear inequalities in two variables
What do we already know about linear inequalities???!!!
HW 364 #’s 2,3,4, 5-8, 10 &11
2. Yes
6. y= 3x +1
3. Yes
4. No
5. y≤ -x + 0
6.
7.
8.
10. y < 3
11. y ≥ x+5
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
1< -2 +4
1 < 2 Yes
y≤x+1
1≤2+1
1 ≤ 3 Yes Satisfies BOTH inequalities
So point (2,1) is a solution!
Independent Practice
Page 370 #’s 2-14 HW
Small group: Maddie, Christian, Nikki work with
Ms. Fisher
Extension: Alex & James HW & 23-28
Review for Chapter 5 Test !!
Page 375 #’s 1-20