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:

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 x 2 + 4x + 5 for x = 3. Solution: x 2 + 4x + 5 {Given expression} = (3) + 5 {Substitute value} = 9 + 4(3) + 5 {Evaluate exponent} = {Do multiplication} = 26 {Addition, left to right}

Another Example Evaluate when x = 3 and y = 4.

Practice Example Evaluate –x 2 + 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.

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 x y  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  = x = 33 Check: x  9 =  9 = = 24 (true)

Solving Equations Adding a NEGATIVE to both sides. x + 12 = 31. x + 12  12 = 31  12 x = 19 Check: x + 12 = = = 31 (true)

Solving Equations Using Multiplication Property of Equality (Multiply both sides by same thing) Solve for x:

Solving Equations Using Multiplication Property of Equality (Dividing both sides by same thing) Solve for 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 {Given Equation} {Collect numbers to right} {Combine LIKE terms} {Divide both sides by 3}

Additional Example {Given Equation} {Apply distributive property} {Combine LIKE terms} {Collect numbers, letters} {Combine LIKE terms} {Divide both sides by 3}

Equations with NO solution {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 {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 x = x x