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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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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



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


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)


State the Inverse Operation

  • Add 7

  • Subtract 3

  • Add -11

  • Subtract -2


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





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 + 2 x – (-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 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 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.






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


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!


Example 1

Solving a Linear Equations

ü

Solve the equations.


Example 1

Solving a Linear Equations

ü

Solve the equations.


Example 2

Combining Like Terms First

ü

Solve the equations.


Example 3

Using the Distributive Property

ü

Solve the equations.


Example 4

Multiplying by a Reciprocal First

ü

Solve the 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


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


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!


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


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


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


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



Example 2

Identify the Number of Solutions

ü

Solve the equations.

one solution

x = -10


Example 2

Identify the Number of Solutions

ü

Solve the equations.

infinite solutions


Example 2

Identify the Number of Solutions

ü

Solve the equations.

no solution


Example 2

Identify the Number of Solutions

ü

Solve the equations.

infinite solutions


Example 2

Identify the Number of Solutions

ü

Solve the equations.

no solution


Example 2

Identify the Number of Solutions

ü

Solve the equations.

one solution

x = 0


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





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


3.6

Solving Equations with Decimals


3.4

Warm-Up

Solve the 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


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


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?


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


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


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


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


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


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


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


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


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 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.


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



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