Chapter 3 Unit Question How do we Solve Equations in Algebra?

Chapter 3

Unit Question

How do we Solve Equations in Algebra?

Open Learning Logs

Date on Left…Section 3 – 2 on right

Warm – Up

1. The reciprocal of -8 is ________.

2. The product of zero and any number is ________.

3. The only number without a reciprocal is ________.

4. There are two numbers that equal their own reciprocal. They are ________ and ________.

Section 2

How do we solve and check equations of the form ax = b?

How do zero and -1 as values for a and b affect the solution?

Homework Check

Multiplication Property of Equality

For all Real numbers m, n, and c…

If m = n, then cm = cn

This is saying that I can multiply an equality by anything,

as long as I multiply BOTH SIDES by the SAME THING

We can expand this to solve equations!

Example: 3x = 6

I choose to let c = 1/3

If we assume this is true then from the general form ax = b we

can say a = 3 and b = 6.

We need to find a value to multiply both sides by to get x

all by itself.3

163

3

1 x

21 x

2x

Summarizing…

So…the trick is coming up with a something to multiply both sides of an equation.

To solve ax = b for x, multiply both sides of the equation by the reciprocal of a!

AND THAT TRICK IS…

Again…solve…

63 x

21 x

2x

3 3A different way…

Something we’ll call…

“Bring down…Bring over”

Use with anything that is NOT a fraction

Example: 3x = 6

Again…solve…

1176 y

5.19y

6 6z20128

z 4.6

20 20

Why not just use “Bring down…Bring Over”

all the time?

Examples: 6

3

1x

3

16 x

…for 𝑙

Solve…

lw

A

w w

𝐴=𝑙∙𝑤

𝐴=𝑙∙𝑤

Solve!

0x = 4 0x = 0

What can you possibly multiply by zero and get 4?

What can you possibly multiply by zero and get 0?

NOTHING!

We call that “No Solution”

EVERYTHING!

We call that “All Real Numbers”

SOLVE!

63

1063

63

1 x

063 x

0x

In set notation then…

{ 0 }The solution is

zero63x = 0

Set of all Real numbers

All Real numbers are solutions

0x = 0

{ } or øThere is no

solution0x = 4

Solution SetSentenceEquation

Solve!

-x = 2.9876

But –x = -1 • x does it not? Substitute!

We need x NOT –x !!!!

-1 • x = 2.9876

-1 • -1 • x = 2.9876 • -1

x = -2.9876

Let’s call this the “Gush” property so we can avoid showing this work!

Solve!

-a = -257.98 0x = 42

0z = 0423y = 0423423

y = 0

{ }

All Real Numbers

a = 257.98

GUSH! GUSH!

Homework

• Do HoffmaSheet 3 – 2

Solve…

When Dax types essays, he types 250 words per minute. About how many minutes will he

need for a 1500 word essay?

Let x = # of minutes 250x = 1500

1500250 x

6x minutes

250 250

Solve…

Suppose a calculator has 6 keys in a row. How many rows are needed for a 56-key

calculator?

Let r = # of rows 6r = 56

566 r

33.9r

6 6

So, 10 rows

Warm - Up

Multiple Choice. For 1 – 5, tell which answer is the solution to the equation.

1. 13x = 52 a. x = ¼ b. x = -4 c. x = 42. -4.5y = 36 a. y = 8 b. y = -8 c.

3. a. z = b. z = 3c. z = 4. a. b. c. 5. 5 = -5m a. b. m = -1 c. m = 0