Algebra 1 Unit 2A Notes: Reasoning with Linear Equations ...
[Pages:38]Algebra 1
Unit 2A: Equations & Inequalities
Notes
Name: ______________________ Block: __________ Teacher: _______________
Algebra 1
Unit 2A Notes: Reasoning with Linear
Equations and Inequalities
DISCLAIMER: We will be using this note packet for Unit 2A. You will be responsible for bringing this packet to class EVERYDAY. If you lose it, you will have to print another one yourself. An electronic copy of this packet can be found on my class blog.
1
Algebra 1
Unit 2A: Equations & Inequalities
Standard
MGSE9-12.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions, quadratic, simple rational, and exponential functions (integer inputs only).
MGSE9-12.A.CED.2 Create linear, quadratic, and exponential equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
MGSE9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret data points as possible (i.e. a solution) or not possible (i.e. a non-solution) under the established constraints.
MGSE9-12.A.CED.4 Rearrange formulas to highlight a quantity of interest using the same reasoning as in solving equations.
MGSE9-12.A.REI.1 Using algebraic properties and the properties of real numbers, justify the steps of a simple, one-solution equation. Students should justify their own steps, or if given two or more steps of an equation, explain the progression from one step to the next using properties.
MGSE9-12.A.REI.3 Solve linear equations and inequalities in one variable including equations with coefficients represented by letters. For example, given ax + 3 = 7, solve for x.
MGSE9-12.A.REI.5 Show and explain why the elimination method works to solve a system of two-variable equations.
MGSE9-12.A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
MGSE9-12.A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.
MGSE9-12.A.REI.11 Using graphs, tables, or successive approximations, show that the solution to the equation f(x) = g(x) is the x-value where the y-values of f(x) and g(x) are the same.
MGSE9-12.A.REI.12 Graph the solution set to a linear inequality in two variables
Notes
Lesson
2
Algebra 1
Unit 2A: Equations & Inequalities
Notes
Unit 2A:Equations & Inequalities
After completion of this unit, you will be able to...
Table of Contents
Learning Target #1: Creating and Solving Linear Equations ? Solve one, two, and multi-step equations (variables on both sides) ? Justify the steps for solving a linear equation ? Create and solve an equation from a context
Learning Target #2: Creating and Solving Linear Inequalities ? Solve and graph a linear inequality ? Create and solve an inequality from a context
Lesson
Day 1 ? Solve 1, 2 & MultiStep Equations
Day 2 ? Solving Equations with Fractions & Justifying Solving
Day 3 ? Graphing & Solving Inequalities
Page 4 7
10
Learning Target #3: Isolating a Variable ? Solve a literal equation (multiple variables) for a specified variable ? Use a Formula to Solve Problems
Learning Target #4: Creating and Solving Systems of Equations ? Identify the solution to a system from a graph or table ? Graph systems of equations ? Determine solutions to a system of equations ? Use a graphing calculator to solve a system of equations ? Use substitution & elimination to solve a system of equations ? Determine the best method for solving a system of equations ? Apply systems to real world contexts
Day 4 ? Creating Equations 13
& Inequalities
Day 5 ? Solving For a
15
Variable
Day 6 ? Graphing Systems 18
of Equations
Day 7 ? Solving Systems by 21
Substitution
Day 8 ? Solving Systems by 23
Elimination
Learning Target #5: Creating and Solving Systems of Inequalities ? Graph linear inequalities & systems of linear inequalities ? Create a linear inequality or system of inequalities from a graph ? Determine the solution to a linear inequality or system of inequalities ? Determine if a given solution is a solution to an inequality or system of inequalities ? Apply inequalities to real world contexts
Day 9 ? Real World
26
Applications of Systems
Day 10 ? More Real World 29
Applications
Day 11 ? Graphing Linear
32
Inequalities
Day 12 ? Graphing
36
Systems of Inequalities &
Applications
Monday August 19th
26th Day 3 ? Graphing & Solving Inequalities
2nd No School ? Labor Day
9th Day 11 ? Graphing Linear Inequalities
Tuesday 20th
27th Day 4 ? Creating Equations & Inequalities from a Context
3rd Day 7 ? Solving Systems
of Equations by Substitution
10th Day 12 ? Graphing
Systems of Inequalities
Wednesday 21st
28th Early Release Day Mixed Practice ? Solving Equations & Inequalities
4th Day 8 ? Solving Systems
of Equations by Elimination
11th Unit 2A Test Review
Thursday 22nd
Day 1 ? Solve 1, 2 & Multi-Step Equations
29th Day 5 ? Solving For a Variable Quiz on Creating & Solving Equations and Inequalities
5th Day 9 ? Real World
Applications Systems of Equations
12th Unit 2A Test
Friday 23rd
Day 2 ? Solving Equations with Fractions & Justifying
Solving 30th Day 6 ? Graphing Systems of Equations
6th Day 10 ? More Real World Applications Systems of Equations Quiz
13th
3
Algebra 1
Unit 2A: Equations & Inequalities Day 1 ? Solving One & Two Step Equations
Notes
Standard(s): MGSE9-12.A.REI.3 Solve linear equations and inequalities in one variable including equations with coefficients represented by letters. For example, given ax + 3 = 7, solve for x.
Expression: ? A mathematical "phrase" composed of terms, coefficients, and variables that stands for a single number, such as 3x + 1 or x2 ? 1. ? We use Properties of Operations to simplify algebraic expressions. Expressions do NOT contain equal signs.
Equation: ? A mathematical "sentence" that says two expressions are equal to each other such as 3x + 1 = 5. ? We use Properties of Equality (inverse operations) to solve algebraic equations. ? Equations contain equal signs.
When solving equations, you must perform inverse operations, which means you have to perform the operation opposite of what you see. You must also remember the operation you perform on one side of the equation must be performed to the other side.
Informal
Operation
Inverse
Addition
Subtraction
Multiplication
Division
Property
Addition Property of Equality
Subtraction Property of Equality
Multiplication Property of Equality
Division Property of Equality
Formal
General Example
If a = b, then a + c = b + c
If a = b, then a ? c = b - c
If a = b, then ac = bc
If a = b, then
Example 1 If x ? 4 = 8, then x = 12 If x + 5 = 7, then x = 2
If , then x = 18
If 2x = 10, then x = 5
Additive Inverse
Multiplicative Inverse (Reciprocal)
A number plus its inverse equals 0.
A number times its reciprocal equals 1.
a + -a = 0 a =1
7 + -7 = 0 3 =1
4
Algebra 1
Unit 2A: Equations & Inequalities
Notes
Solving One Step Equations Practice
Practice: Solve each equation.
1.
x ? 4 = 3
Operation You See: _______________
Inverse Operation: _______________
2.
y + 4 = 3
3.
s =9
3
4.
6p = 12
Operation You See: _______________ Operation You See: _______________ Operation You See: _______________
Inverse Operation: _______________ Inverse Operation: _______________ Inverse Operation: _______________
Practice: Solve each equation on your own.
a.
x ? 6 = 10
b.
-5d = 25
c.
8 + m = -4
d.
x =1
7
e.
y ? (-9) = 2
f.
Solving Two Step Equations
When solving equations with more than one step, you still want to think about how you can "undo" the operations you see.
Practice: Solve each equation, showing all steps, for each variable.
1. 3x - 4 = 14
2. 2x + 4 = 10
3. 7 ? 3y = 22
4. 0.5m ? 1 = 8
5. -6 + = -5
6. x - 8 = -5 4
5
Algebra 1
Unit 2A: Equations & Inequalities Solving Multi-Step Equations
Notes
Multi-step equations mean you might have to add, subtract, multiply, or divide all in one problem to isolate the variable. When solving multi-step equations, you are using inverse operations, which is like doing PEMDAS in reverse order.
Multi - Step Equations with Combining Like Terms
Practice: Solve each equation, showing all steps, for each variable.
a. -5n + 6n + 15 ? 3n = -3
b. 3x + 12x ? 20 = 25
c. -2x + 4x ? 12 = 40
Multi - Step Equations with the Distributive Property
Practice: Solve each equation, showing all steps, for each variable.
a. 2(n + 5) = -2
b. 4(2x ? 7) + 5 = -39
c. 6x ? (3x + 8) = 16
Multi ? Step Equations with Variables on Both Sides
Practice: Solve each equation, showing all steps, for each variable
a. 5p ? 14 = 8p + 4
b. 8x ? 1 = 23 ? 4x
c. 5x + 34 = -2(1 ? 7x)
6
Algebra 1
Unit 2A: Equations & Inequalities Day 2 - Equations with Fractions and Decimals
Notes
Standard(s): MGSE9-12.A.REI.3 Solve linear equations and inequalities in one variable including equations with coefficients represented by letters. For example, given ax + 3 = 7, solve for x.
MGSE9-12.A.REI.1 Using algebraic properties and the properties of real numbers, justify the steps of a simple, one-solution equation. Students should justify their own steps, or if given two or more steps of an equation, explain the progression from one step to the next using properties.
When you solve equations with decimals, you solve them as if you would an equation without decimals.
1. 3.5x ? 37.9 = .2x
2. 14.7 + 2.3x = 4.06
3. -1.6 ? 0.9w = 11.6 + 2.4w
Equations with Fractions
When solving equations with fractions, you want to find a way to eliminate the fraction.
To eliminate the fraction, multiply by a Common Denominator
1. - 2 m = 10 3
2.
3x = 6
4
3.
- 3 x -1= 8
2
4.
2m + 5 = 12
3
1.
w + 1 = 6w -1
77
2.
x + 2x = 5
63
3. x + 3 - x = 5 82
7
Algebra 1
Unit 2A: Equations & Inequalities
Special Types of Solutions
Solve the following equations. What do you notice about the solutions?
a. 2x ? 7 + 3x = 4x + 2
b. 3(x ? 5) + 11 = x + 2(x + 5)
Notes
c. 3x + 7 = 5x + 2(3 ? x) + 1
Justifying the Solving of Equations
Property Commutative Property of
Addition Associative Property of
Addition
Additive Identity
Additive Inverse
Commutative Property of Multiplication
Associative Property of Multiplication
Multiplicative Identity Multiplicative Inverse
(Reciprocal) Zero Property of
Multiplication
Distributive Property
Properties of Addition Operations
What It Means
General Example
Rearrange the order and the sum will stay the same.
Change the order of the grouping and the sum will
stay the same. Zero added to any number will equal that
number. A number plus its inverse
equals 0.
a + b = b + a (a + b) + c = a + (b + c)
a + 0 = a a + -a = 0
Properties of Multiplication Operations
Rearrange the order and the product will stay the
same. Change the order of the grouping and the product
will stay the same. One times any number
equals that number.
a b = b a (a b) c = a (b c)
a 1 = a
A number times its reciprocal equals 1.
a =1
Any number times 0 will always equal 0.
a 0 = 0
Multiply a number to every term within a quantity (parenthesis).
a(b + c) = ab + ac
Example 1 2 + 4 = 4 + 2
(4 + 6) + 1 = 4 + (6 + 1)
4 + 0 = 4 7 + -7 = 0
5 2 = 2 5
(3 4) 2 = 3 (4 2) 8 1 = 8 3 =1 7 0 = 0
4(x + 5) = 4x + 4(5) = 4x + 20
8
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