MM150 Unit 3 Seminar Agenda Seminar Topics Order of Operations Linear Equations in One Variable...

MM150 Unit 3 Seminar Agenda

Seminar Topics

• Order of Operations

• Linear Equations in One Variable

• Formulas

• Applications of Linear Equations

Definitions

• Algebra: a generalized form of arithmetic.• Variables: letters used to represent numbers• Constant: symbol that represents a specific

quantity• Algebraic expression: a collection of variables,

numbers, parentheses, and operation symbols.

Examples:

24 2, 4, 4(3 5), , 8 2

3 5

xx x y y y

x

Order of Operations

1. First, perform all operations within parentheses or other grouping symbols (according to the following order).

2. Next, perform all exponential operations (that is, raising to powers or finding roots).

3. Next, perform all multiplication and divisions from left to right.

4. Finally, perform all additions and subtractions from left to right.

Sometimes abbreviated PEMDAS

Example, Order of Operations

Evaluate the expression

x2 + 4x + 5 for x = 3.

Solution: x2 + 4x + 5 {Given expression}

= 32 + 4(3) + 5 {Substitute value}

= 9 + 4(3) + 5 {Evaluate exponent}

= 9 + 12 + 5 {Do multiplication}

= 26 {Addition, left to right}

Another Example

• Evaluate when x = 3 and y = 4. 2 24 3 5x xy y

2 2

2 2

4(3) 3(3)(4) 5(4 )

4(9) 36 5(16)

36 36 80

0 80

8

4 3 5

0

x xy y

Practice Example

Evaluate –x2 + 5xy, when x = 2 and y = -3.

More Definitions

• Terms are parts that are added or subtracted in an algebraic expression.

• Coefficient is the numerical part of a term.• Like terms are terms that have the same variables

with the same exponents on the variables.

• Unlike terms have different variables or different exponents on the variables.

2 22 , 7 5 , 8x x x x

3 22 , 7 5 , 6x x x

Properties of REAL Numbers

Associative property of multiplication

(ab)c = a(bc)

Associative property of addition

(a + b) + c = a + (b + c)

Commutative property of multiplication

ab = ba

Commutative property of addition

a + b = b + a

Distributive propertya(b + c) = ab + ac

Combine LIKE terms

• 8x + 4x= (8 + 4)x= 12x

• 5y 6y= (5 6)y= y

• x + 15 5x + 9

= (1 5)x + (15 + 9)

= 4x + 24

• 3x + 2 + 6y 4 + 7x

= (3 + 7)x + 6y + (2 4)

= 10x + 6y 2

Practice Combining LIKE terms

2x – 4y 3x + 4y -15

=

Solving Equations

Using ADDITION property of Equality:

(Add same thing to both sides)

x 9 = 24.

x 9 + 9 = 24 + 9

x = 33

Check: x 9 = 24

33 9 = 24

24 = 24 (true)

Solving Equations

Adding a NEGATIVE to both sides.

x + 12 = 31. x + 12 12 = 31 12

x = 19 Check: x + 12 = 31

19 + 12 = 31 31 = 31 (true)

Solving Equations

Using Multiplication Property of Equality (Multiply both sides by same thing)

Solve for x:

x

79

7x

7

7(9)

1 7 x1 7

63

x 63

Solving Equations

Using Multiplication Property of Equality (Dividing both sides by same thing)

Solve for x:

4

3 4 17

3 4 17

3 21

3 21

4

3 37

x

x

x

x

x

General steps for Solving Equations

• If the equation contains fractions, multiply both sides of the equation by the lowest common denominator (or least common multiple). This will eliminate fractions.

• Use the distributive property to remove parentheses.

• Combine like terms on the same side of the equal sign when possible.

• Use the addition or subtraction property to collect all terms with a variable on one side of the equal sign and all constants on the other side of the equal sign.

• Solve for the variable using the division or multiplication property.

Solving more Complex Equations

• Solve 3x 4 = 17

4

3 4 17

3 4 17

3 21

3 21

4

3 37

x

x

x

x

x

{Given Equation}

{Collect numbers to right}

{Combine LIKE terms}

{Divide both sides by 3}

Additional Example

21 6 3( 2)

21 6 3 6

21 3 12

1221 3 12

9 3

9 3

3

12

3 3

x

x

x

x

x

x

x

{Given Equation}

{Apply distributive property}

{Combine LIKE terms}

{Collect numbers, letters}

{Combine LIKE terms}

{Divide both sides by 3}

Equations with NO solution

6( 2) 2 3 4(2 3) 2

6 12 2 3 8 12 2

8 9 8 10

8 8 9 8 8 10

9 10

x x x

x x x

x x

x x x x

{Given Equation}

{Apply distributive property}

{Combine LIKE terms}

{Collect variables, numbers}

{Combine like terms}

False, the equation has no solution. The equation is inconsistent.

Equations with Infinite Solutions

4( 1) 6( 2) 2( 4)

4 4 6 12 2 8

2 8 2 8

2 2 8 2 2 8

8 8

8 8 8 8

0 0

x x x

x x x

x x

x x x x

{Given Equation}

{Apply distributive property}

{Combine LIKE terms}

{Collect variables, numbers}

{Combine like terms}

True statement, so the solution is ALL real numbers.

Equation containing Fractions

Variables on BOTH sides

6x + 8 = 10x + 12

Practice Problems

(x – 5) = (x – 9) 4 3

Practice Problems

Combine like terms:

9 (x – 3) – 2 (x + 5) + 10

Practice Problems

Solve for x:

6x + 8 - 22x = 28 + 14x - 10 + 12x