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Chapter 3 – Solving Linear Equations

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Chapter 3 – Solving Linear Equations

Algebra 1

Fall 2013

- 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

- Solve linear equations using addition and subtraction
- Use linear equations to solve a variety of real-life problems

- Has an equal sign
- Combination of numbers and variables
- Complete thought with numbers and variables on left and right of the equal sign

- Equations have two sides
- Want to keep both sides equal

“What value does x have?”

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

- We need a process to solve more difficult problems
- We need to use 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

Check

Check

Check

Check

- 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

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

- 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?

- 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?

- 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.

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

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

x + 2 = 11 + 2x – (-2) = 5

-x = -10

- 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

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

This problem means -4 times x.

This problem means x divided by 5.

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

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

- 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!

- 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!

- 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:

- 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

- 3.1-3.3 Quiz on Thursday, Sept. 19th
- Homework:
- P. 157 #’s 18-22

Solving Equations with Variables on Both Sides (Day 2)

infinite solutions

no solution

- 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

- 3.1-3.3 Quiz on Thursday, Sept. 19th
- Homework:
- P. 169 #’s 26-33
- P. 171 # 53

3.7 – Formulas and Functions

- Objective
- Solve a formula for one of its variables

- 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)

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

- We want the equation to say “l equals…”
- Think:

A = l ∙ w

How do I get l by itself?

- Perform the inverse operations to isolate the variable.

A = l ∙ w

l = (A/w)

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

- 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)

- 15x + 5y = 10
- 1 + 7y = 5x – 2
- 7x + 5x = -8 + 2y

- Ch. 3 Test is on Wednesday, Sept. 25th
- Homework:
- P. 177 #’s 13-14

What is represented by this brain puzzle?

H I J K L M N O