Chapter 3 Solving Equations Introduction to Equations Equation: equality of two mathematical expressions. = 9 + 3 = 12 3x – 2 = 10 y² + 4 = 2y - 1

Solution to an equation, is the number when substituted for the variable makes the equation a true statement. Is –2 a solution or 2x + 5 = x² - 3 ? Substitute –2 in for the x 2(-2) + 5 = (-2)² - 3 -4 + 5 = 4 - 3 1 = 1

Solve an equation Addition Property r – 6 = 14 We use the Addition + 6 +6 method by adding positive 6 to both sides of the equation. r = 20 *CHECK your solution

Solve an equation s + ¾ = ½ - ¾ -¾ Using the Addition Method add a negative ¾ to both sides. s = -¼ Remember to get a common denominator. Check your solution.

Solving Equations 3y = 27 Using the Multiplication Method we divide by the coefficient, which is the same as multiplying by ⅓ 3y = 27 3 3 y = 9 Check your solution

Solving Equations Using the multiplication method we multiply the reciprocal of the coefficient to both sides. X = 10 Check 4/5(10) = 8 8 = 8

Solving Equations: 2 Step 6x + 12 = 36 6x + 12 = 36 - 12 -12 Addition Method 6x = 24 6x = 24 Multiplication Method 6 6 x = 4 Check

Basic Percent Equations Percent • Base = Amount P • B = A 20% of what number is 30 multiply B equals .2 • B = 30 B = 150

Basic Percent Equations Percent • Base = Amount P • B = A What Percent of 80 is 70 P multiply equals P • 80 = 70 P = .875 P = 87.5% Convert to percentage.

Basic Percent Equations Percent • Base = Amount P • B = A 25% of 60 is what? multiply equals amount .25 • 60 = A 15 = A

Steps to solve equations: 1. Remove all grouping symbols Look to collect the left side and the right side. Add the opposite of the smallest variable term to each side. Add the opposite of the constant that’s on the same side as the variable term to each side.

Steps to solve equations continued 5. Divide by the coefficient. *variable term = constant term *if the coefficient is a fraction, multiply by the reciprocal. 6. CHECK the solution.

Ex. Solving Equations 3x – 4(2 – x) = 3(x – 2) - 4 3x – 8 + 4x = 3x – 6 – 4 Distribute 7x – 8 = 3x - 10 Collect like terms -3x -3x 4x – 8 = -10 Add opposite of the Smallest variable term + 8 + 8 4x = -2 Add the opposite of the constant 4 4 Divide by the Coefficient. x = -½

Ex. 2 Solving Equations -2[4 – (3b + 2)] = 5 – 2(3b + 6) -2[4 – 3b – 2] = 5 – 6b - 12 -8 + 6b + 4 = 5 – 6b - 12 6b – 4 = -6b - 7 Collected 12b – 4 = -7 Added 6b 12b = -3 Added 4 b = ¼ Divided by 12 and reduced CHECK

Translating Sentences into Equations Equation-equality of two mathematical expressions. Key words that mean = equals is is equal to amounts to represents

Ex. Translate: “five less than a number is thirteen” n - 5 = 13 n = 18 Solve

Translate Consecutive Integers Consecutive integers are integers that follow one another in order. Consecutive odd integers- 5,7,9 Consecutive even integers- 8,10,12

CHAPTER 4 POLYNOMIALS Polynomial: a variable expression in which the terms are monomials. Monomial: one term polynomial 5, 5x², ¾x, 6x²y³ Not: or 3 Binomial: two term polynomial 5x² + 7 Trinomial: Three term polynomial 3x² - 5x + 8

Addition and Subtraction Polynomials can be added vertically or horizontally. Horizontal Format Collect like terms Ex. ( 3x³ - 7x + 2) + (7x² + 2x – 7) 3x³ + 7x² - 5x - 5

Addition and Subtraction Vertical Format Ex. ( 3x³ - 7x + 2) + (7x² + 2x – 7) ³ ² ¹ º 3x³ - 7x + 2 +7x² + 2x – 7 Organized in columns by the degree 3x³+7x² - 5x - 5

Subtraction Horizontal Format (-4w³ + 8w – 8) – (3w³ - 4w² - 2w – 1) Change subtraction to addition of the opposite (-4w³ + 8w – 8)+(-3w³ + 4w² + 2w + 1) -7w³ + 4w² + 10w - 7

Subtraction Vertical Format (-4w³ + 8w – 8) – (3w³ - 4w² - 2w – 1) Change subtraction to the addition of the opposite ³ ² ¹ º -4w³ + 8w - 8 -3w³ + 4w² +2w + 1 -7w³ - 4w² + 6w - 9

Multiplication of Monomials Remember x³ = x • x • x & x² = x • x Then x³ • x² = x • x • x • x • x = x5 RULE 1 xn • xm = x n+m when multiplying similar bases add the powers. Ex. y4 • y • y3 = y 4+1+3 = y8

Multiplying Monomials Ex. (8m³n)(-3n5) *Multiply the coefficients, *Multiply similar bases by adding the powers together -24m3n6

Simplify powers of Monomials (x4)3 = x4 • x4 • x4 = x4 + 4 + 4 = x12 Rule 2 (x m)n = xmn Multiply the outside power with the power on the inside. Rule 3 (xmyn)p = xmpynp Ex. (5x²y³)³ = 51•3x2•3y3•3 = 125x6y9

Simplify Monomials Continue Ex. (ab²)(-2a²b)³ Rule 3: Multiply the outside power to inside powers. (ab²)(-8a6b³) -8a7b5 Rule 1: multiply the Monomials by adding the exponents

Multiplication of Polynomials Distribute and follow Rule 1 -3a(4a² - 5a + 6) -12a³ + 15a² - 18a

Multiplication of two Polynomials *when multiplying two polynomials you will use Distributive Property. *be sure every term in one parenthesis is multiplied to every term in the other parenthesis.

Multiplication of two Polynomials Ex. (y – 2)(y² + 3y + 1) Multiply y to every term. Multiply –2 to y³ + 3y² + y - 2y² - 6y - 2 Combine like terms. y³ + y² - 5y - 2

Multiply two Binomials The product of two binomials can be found using the FOIL method. F First terms in each parenthesis. O Outer terms in each parenthesis. I Inner terms in each parenthesis. L Last terms in each parenthesis.

Multiply two Binomials Ex. (2x + 3)(x + 5) O F F O I L (2x + 3)(x + 5) = 2x² +10x +3x +15 I L Collect Like Terms 2x² + 13x + 15

Special Products of Binomials Sum and Difference of two Binomials (a + b)(a – b) Square the first term Square the second term a² - b² Minus sign between the products

Sum and Difference of Binomials (2x + 3)(2x – 3) Square the term 2x - 4x² 9 Square the term 3 Minus sign between the terms

Or use the short cut Square of a Binomial (a + b)² = (a + b)(a + b) Then FOIL Or use the short cut (a + b)² = a² + 2ab + b² 1. Square 1st term 2. Multiply terms and times by 2. ab times 2 = 3. Square 2nd term

Square of a Binomial Ex. (5x + 3)² = 25x² + 30x + 9 Square 5x Multiply 5x and 3 then times by 2 Square the 3

Square of a Binomial (4y – 7)² = 16y² - 56y + 49 Square the 4y Mutiply the 4y and 7 then times by 2 Square the 7

Integer Exponents Divide Monomials x•x•x•x•x x•x x³ = = Rule 4 xm xn When m > n Xm-n = xm xn 1 xn-m When n > m =

Integer Exponents Divide Monomials r8t6 r5t r8-5t6-1 = r3t5 = 1 a6-4b9-7 1 a²b² a4b7 a6b9 = = a5b3c8d4 a2b7c4d9 a5-2c8-4 b7-3d9-4 a3c4 b4d5 = =

Integer Exponents Zero and Negative Exponents Rule 5 a0 = 1 a ≠ 0 x³ = x3-3 = xº *Summary any number (except for 0) or variable raised to the power of zero = 1

Integer Exponents Zero and Negative Exponents x-n = 1 xn Rule 6: 1 X-n = xn and 1 1 If we make everything a fraction, we can see that we take the base and it’s negative exponent and move them from the numerator to the denominator and the sign of the exponent changes.

Integer Exponents Zero and Negative Exponents 2 5a-4 = 2a4 5