Chapter 3 solving linear equations
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Chapter 3 – Solving Linear Equations. Algebra 1 Fall 2013. What will we do in Chapter 3?. Solve linear equations using addition and subtraction Use linear equations to solve real-life problems Use multiplication and division to solve linear equations

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Chapter 3 – Solving Linear Equations

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Chapter 3 solving linear equations

Chapter 3 – Solving Linear Equations

Algebra 1

Fall 2013


What will we do in chapter 3

What will we do in Chapter 3?

  • Solve linear equations using addition and subtraction

  • Use linear equations to solve real-life problems

  • Use multiplication and division to solve linear equations

  • Solve multi-step equations with variables on both sides of the equals sign

  • Problem solve with various equations

  • Learn how to use manipulate decimals in linear equations


3 1 solving equations using addition and subtraction

3.1 – Solving Equations Using Addition and Subtraction


Objectives

Objectives

  • Solve linear equations using addition and subtraction

  • Use linear equations to solve a variety of real-life problems


What is an equation

What is an Equation?

  • Has an equal sign

  • Combination of numbers and variables

  • Complete thought with numbers and variables on left and right of the equal sign


Equation scale

Equation = Scale

  • Equations have two sides

  • Want to keep both sides equal


Chapter 3 solving linear equations

x = 2

“What value does x have?”


X 3 5

x + 3 = 5

We need x to have a value that will balance the scale!


What if mental math isn t an option

What if Mental Math isn’t an Option?

  • We need a process to solve more difficult problems

  • We need to use inverse operations


Inverse operations

Inverse Operations

  • Definition:

  • Examples:

  • Why do we use inverse operations?

Operations that undo each other

Addition and subtraction

Multiplication and division

To isolate the variable (get the variable by itself)


Chapter 3 solving linear equations

State the Inverse Operation

  • Add 7

  • Subtract 3

  • Add -11

  • Subtract -2


Chapter 3 solving linear equations

Steps to Solving One-Step Equations

  • Simplify both sides of the equation

  • Isolate the variable (by using the inverse operation)

  • Find the solution

  • Check your solution


X 5 16

x - 5 = 16

Check


X 5 13

x - 5 = -13

Check


8 x 4

-8 = x – (-4)

Check


X 5 2 10

x - 5 + 2 = -10

Check


Linear equation

Linear Equation

  • Definition:

Equation in which the variable is raised to the first power and does not occur in a denominator inside a square root symbol, or inside absolute value symbols


Find the linear equations

Find the Linear Equations

x + 5 = 9

x2 – 8 = 16

-4 + x = 7 – 3x

16 + 5 = (x/5)

(2/x) + 1 = -10

| -17 + x | = 1

x3 + (-3) = 12

-6 = x


Writing equations

Writing Equations

  • You have x dollars and your friend pays you $6 that he owed you. You now have $14. How much money did you have before your friend paid you?


Writing equations1

Writing Equations

  • A telephone pole extends 4 feet below the ground and 16 feet above the ground. What is the total length x of the telephone pole?


Equivalent equations

Equivalent Equations

  • Definition:

Two equations that have the same solution

2 + x = 9

x + (-2) = 5

The equations are equivalent because the solution to both equations is x = 7.


Find the equivalent equations

Find the Equivalent Equations

Draw an arrow from the equations on the left to their equivalent equations on the right.

x + 1 = 9

x = -7

8 + x = 5

x + (-3) = -10

4 + (-7) = x

5 + x = 13


Tricky equations

Tricky Equations

| -6 | + x = 11| 5 | + 7 = x

x + 2 = 11 + 2x – (-2) = 5

-x = -10


Reminders

Reminders

  • 3.1-3.3 Quiz on Thursday, Sept. 12th

  • Homework:

    • P. 135 #’s 33-44

    • P. 137 #’s 66-68


3 2 solving equations using multiplication and division

3.2 – Solving Equations Using Multiplication and Division


3 2 solving equations using multiplication and division1

3.2 – Solving Equations Using Multiplication and Division

  • Objective: Solve one-step equations using multiplication and division.


State the inverse operation

State the inverse operation.

This problem means -4 times x.


State the inverse operation1

State the inverse operation.

This problem means x divided by 5.


Solve the equation

Solve the equation.


Solve the equation1

Solve the equation.


Solve the equation2

Solve the equation.


Solve the equation3

Solve the equation.


Solve the equation4

Solve the equation.

This problem means (-2/3) times m equals 10.


Remember

REMEMBER!

  • Dividing by a fraction is the same as multiplying by the reciprocal.


Solve the equation5

Solve the equation.


Solve the equation6

Solve the equation.


Solve the equation7

Solve the equation.


Solve the equation8

Solve the equation.


Word problems

Word Problems

  • You ate three of the eight slices of pizza and you paid $3.30 as your share of the cost. How much did the whole pizza cost? Write an equation!


Word problems1

Word Problems

  • Each household receives about 676 pieces of junk mail per year. About how many pieces of junk mail does a household receive per week? Write an equation!


Reminders1

Reminders

  • 3.1-3.3 Quiz on Thursday, Sept. 12th

  • Homework:

    • P. 142 #’s 28-36, 49-50


Chapter 3 solving linear equations

3.3

Solving Multi-Step Equations

When solving equations you must balance both side.

  • Simplify both sides of the equation.

  • Do the opposite operation to both sides.

    • Addition/Subraction

    • Multiplication/Division

  • Check!


Chapter 3 solving linear equations

Example 1

Solving a Linear Equations

ü

Solve the equations.


Chapter 3 solving linear equations

Example 1

Solving a Linear Equations

ü

Solve the equations.


Chapter 3 solving linear equations

Example 2

Combining Like Terms First

ü

Solve the equations.


Chapter 3 solving linear equations

Example 3

Using the Distributive Property

ü

Solve the equations.


Chapter 3 solving linear equations

Example 4

Multiplying by a Reciprocal First

ü

Solve the equations.


Chapter 3 solving linear equations

Example 5

Real World: Solving Equations

A body temperature of 95°F or lower may indicate the medical condition called hypothermia. What temperature in the Celsius scale may indicate hypothermia? Use the formula:


Reminders2

Reminders

  • 3.1-3.3 Quiz on Thursday, Sept. 12th

  • Homework:

    • P. 148-149 #’s 16-36 EVEN’S ONLY


Chapter 3 solving linear equations

Solving Equations with Variables on Both Sides

3.4

Objectives:

  • Solve equations with variables on both sides.

  • Solve equations with variables in the real world.

    Vocabulary:

    none


Chapter 3 solving linear equations

Solving Equations with Variables on Both Sides

3.4

When solving equations you must balance both side.

  • Simplify both sides of the equation.

  • Put variable on one side.

  • Do the opposite operation to both sides.

    • Addition/Subtraction

    • Multiplication/Division

  • Check!


Chapter 3 solving linear equations

Example 1

Collect Variables on One Side

  • Simplify both sides of the equation.

  • Put variable on one side.

  • Do the opposite operation to both sides.

    • Addition/Subtraction

    • Multiplication/Division

  • Check!

Solve the equations.

ü

x = 2


Chapter 3 solving linear equations

Example 1

Collect Variables on One Side

  • Simplify both sides of the equation.

  • Put variable on one side.

  • Do the opposite operation to both sides.

    • Addition/Subtraction

    • Multiplication/Division

  • Check!

Solve the equations.

ü

x = -2


Chapter 3 solving linear equations

Example 1

Collect Variables on One Side

  • Simplify both sides of the equation.

  • Put variable on one side.

  • Do the opposite operation to both sides.

    • Addition/Subtraction

    • Multiplication/Division

  • Check!

Solve the equations.

ü

x = 1


Chapter 3 solving linear equations

Example 1

Collect Variables on One Side

  • Simplify both sides of the equation.

  • Put variable on one side.

  • Do the opposite operation to both sides.

    • Addition/Subtraction

    • Multiplication/Division

  • Check!

Solve the equations.

ü

x = 5


Identifying the number of solutions

Identifying the Number of Solutions


Chapter 3 solving linear equations

Example 2

Identify the Number of Solutions

ü

Solve the equations.

one solution

x = -10


Chapter 3 solving linear equations

Example 2

Identify the Number of Solutions

ü

Solve the equations.

infinite solutions


Chapter 3 solving linear equations

Example 2

Identify the Number of Solutions

ü

Solve the equations.

no solution


Chapter 3 solving linear equations

Example 2

Identify the Number of Solutions

ü

Solve the equations.

infinite solutions


Chapter 3 solving linear equations

Example 2

Identify the Number of Solutions

ü

Solve the equations.

no solution


Chapter 3 solving linear equations

Example 2

Identify the Number of Solutions

ü

Solve the equations.

one solution

x = 0


Chapter 3 solving linear equations

Example 2

Identify the Number of Solutions

Determine the number of solutions without solving.

7y + 3 = 7y + 4

6y + 3 = 3 + 6y

3y + 8 = 8 + 4y

10 – 11y = 10 + 11y

15a + 2 = 10a + 3 + 5a

no solutions

infinite solutions

one solution

one solution

no solutions


Reminders3

Reminders

  • 3.1-3.3 Quiz on Thursday, Sept. 19th

  • Homework:

    • P. 157 #’s 18-22


Solving equations with variables on both sides day 2

Solving Equations with Variables on Both Sides (Day 2)


Partner practice

Partner Practice


Partner practice1

Partner Practice


Partner practice2

Partner Practice

infinite solutions


Partner practice3

Partner Practice


Smartpal practice

SmartPal Practice


Partner practice4

Partner Practice

no solution


Reminders4

Reminders

  • 3.1-3.3 Quiz on Thursday, Sept. 19th

  • Homework:

    • P. 157 #’s 24-26, 31-33, 37


Chapter 3 solving linear equations

3.6

Solving Equations with Decimals


Chapter 3 solving linear equations

3.4

Warm-Up

Solve the equations.


Chapter 3 solving linear equations

3.6

Solving Decimal Equations

Objectives:

Solve equations involving decimals.

Apply decimal equations to real-life applications

Vocabulary:

exact answer, approximate answer, percent of


Chapter 3 solving linear equations

3.6

Solving Decimal Equations

Vocabulary:

exact answer – use an =

approximate answer – use an ≈

percent of – change the percent to a decimal and multiply

20% of 32

= 6.4

.20

x

32


Chapter 3 solving linear equations

Example 1

Round for the Final Answer

  • Simplify both sides of the equation.

  • Put variable on 1 side.

  • Do the opposite operation to both sides.

    • Addition/Subtraction

    • Multiplication/Division

  • Check!

Solve the equation.

Round to the nearest hundredth.

ü

x ≈ -1.40

Why did we use the ≈ symbol?


Chapter 3 solving linear equations

Example 1

Round for the Final Answer

  • Simplify both sides of the equation.

  • Put variable on 1 side.

  • Do the opposite operation to both sides.

    • Addition/Subtraction

    • Multiplication/Division

  • Check!

Solve the equation.

Round to the nearest hundredth.

ü

x ≈ -1.12


Chapter 3 solving linear equations

Example 2

Solve Equations with Decimals

  • Simplify both sides of the equation.

  • Put variable on 1 side.

  • Do the opposite operation to both sides.

    • Addition/Subtraction

    • Multiplication/Division

  • Check!

Solve the equation.

Round to the nearest tenth.

ü

x ≈ 5.5


Chapter 3 solving linear equations

Star Game

  • Each student will receive a magnet

  • If you get the problem correct on your first try, you can slide your magnet from star to star

  • DO NOT MOVE ANYONE ELSE’S MAGNET


Chapter 3 solving linear equations

Example 2

Solve Equations with Decimals

  • Simplify both sides of the equation.

  • Put variable on 1 side.

  • Do the opposite operation to both sides.

    • Addition/Subtraction

    • Multiplication/Division

  • Check!

Solve the equation.

Round to the nearest tenth.

ü

y ≈ 29.1


Chapter 3 solving linear equations

Example 3

Rounding for a Practical Answer

  • Simplify both sides of the equation.

  • Put variable on 1 side.

  • Do the opposite operation to both sides.

    • Addition/Subtraction

    • Multiplication/Division

  • Check!

Three people want to share the cost of a pizza equally. The pizza costs $12.89. What should each person pay?

ü

$4.30


Chapter 3 solving linear equations

Example 4

Change Decimals to Integers

  • Simplify both sides of the equation.

  • Put variable on 1 side.

  • Do the opposite operation to both sides.

    • Addition/Subtraction

    • Multiplication/Division

  • Check!

Solve the equation.

Round to the nearest tenth.

ü

n ≈ 5.1


Chapter 3 solving linear equations

Example 4

Change Decimals to Integers

  • Simplify both sides of the equation.

  • Put variable on 1 side.

  • Do the opposite operation to both sides.

    • Addition/Subtraction

    • Multiplication/Division

  • Check!

Solve the equation.

Round to the nearest tenth.

ü

n ≈ 4.0


Chapter 3 solving linear equations

Example 5

Finding Percents of Numbers

  • Find the percents of the prices, rounding to the nearest cent.

  • 5% of $23.45

  • 7% of $62.50

  • 11% of $99.99

  • 15% of $48.28

$1.17

$4.38

$11.00

$7.24


Chapter 3 solving linear equations

Example 6

Problems with Decimals

You buy a baseball cap at the stadium for a total cost of $35.51. This included the 11% sales tax. What was the original cost of the cap?

$31.99


Reminders5

Reminders

  • 3.1-3.3 Quiz on Thursday, Sept. 19th

  • Homework:

    • P. 169 #’s 26-33

    • P. 171 # 53


3 7 formulas and functions

3.7 – Formulas and Functions


3 7 formulas and functions1

3.7 – Formulas and Functions

  • Objective

    • Solve a formula for one of its variables


What is a formula

What is a Formula?

  • Definition:

    • An algebraic __________ that relates two or more ___________ quantities.

  • Examples:

    • Area of rectangle

    • Temperature

equation

real-life

A = l ∙ w

C = 5/9 ∙ (F – 32)


Using the area formula

Using the Area Formula

A = l ∙ w

  • What is the area of a rectangle with…

    • l = 5 cm, w = 12 cm

    • l = 7 ft, w = 9 ft

    • l = 3 cm, w = 13 cm

60 cm2

63 cm2

39 cm2

Notice: We were given a length and width.


Chapter 3 solving linear equations

Using the Area Formula

A = l ∙ w

  • What is the length of a rectangle with an area of 228 cm2 and a width of 12 cm?

    • Turn to your partner and figure out a solution.

Notice: We are given a different set of information in this problem.

l = 19 cm


Try to solve for the length

Try to Solve for the Length

  • We want the equation to say “l equals…”

  • Think:

A = l ∙ w

How do I get l by itself?


Try to solve for the length1

Try to Solve for the Length

  • Perform the inverse operations to isolate the variable.

A = l ∙ w

l = (A/w)


Solving for length

Solving for Length

l = (A/w)

  • Use this new formula to solve for length.

    • A = 49 cm2, w = 7 cm

    • A = 108 ft2, w = 12 ft

7 cm

9 ft


Using the distance formula

Using the Distance Formula

  • Solve the distance formula for time (t).

    • The equation should say “t equals…”

  • Solve the distance formula for the rate (r).

    • The equation should say “r equals…”

d = r ∙ t

t = (d/r)

r = (d/t)


Rewrite the equation so that y is a function of x

Rewrite the equation so that y is a function of x.

  • 15x + 5y = 10

  • 1 + 7y = 5x – 2

  • 7x + 5x = -8 + 2y


Reminders6

Reminders

  • Ch. 3 Test is on Wednesday, Sept. 25th

  • Homework:

    • P. 177 #’s 13-14


Chapter 3 solving linear equations

What is represented by this brain puzzle?

H I J K L M N O


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