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Algebra 1 Chapter 1 Notes Introduction to Algebra

Algebra 1 Chapter 1 Notes Introduction to Algebra. ALGEBRA is the process of moving values from one side of equation to the other without changing the equality. KEEP IT BALANCED !.

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Algebra 1 Chapter 1 Notes Introduction to Algebra

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  1. Algebra 1 Chapter 1 Notes Introduction to Algebra

  2. ALGEBRAis the process of moving values from one side of equation to the other without changing the equality. KEEP IT BALANCED ! If you change one side of an equation, you must change the other side equally. For example, if x = y, then x + 1 = y + 1

  3. 1.1 Algebraic Expressions The Study of Algebra involves numbers and operations. A Numerical Expression contains one of more numbers and one or more operations: 12 7.6 5 + 9 14 – 7 x 2 In Algebra, letters are often used to represent numbers. These letters are called Variables. An Algebraic Expression contains one of more variables and one or more operations: 5n 4n − 6 3y (2) To Evaluate an Expression replace each variable with a number to find a numerical value. Example 1: Evaluate 5n where n = 6, thus So, 5 (6) = 30 Example 2: Evaluate 2xy for x = 4 and y = 3, thus 2 (x) (y) = 2 (4) (3) = 24

  4. 1.2 Order of Operations • Order of Operations • Do all multiplications and divisions in order from left to right • Do all additions and subtractions in order from left to right.

  5. 1.2 Grouping Symbols Grouping Symbols Parentheses ( ) and brackets [ ] are called Grouping Symbols. The rule is to do operations within grouping symbols first. Note: a multiplication symbol may be omitted when it occurs next to a grouping symbol.

  6. 1.3 Exponents Exponent The exponent indicates the number of times the vase is used as a factor. exponent 53 = 5 ● 5 ● 5 = 125 53 base • Order of Operations • Operate within groups symbols first. Work from the inside to the outside. • Simplify powers. • Multiply and divide from left to right. • Add and subtract from left to right. 42 ● 13 + 8 4 ● 4 ● 1 ● 1 ● 1 + 8 16 ● 1 + 8 = 24

  7. 1.3 Exponents and Grouping Symbols The exponent outside a grouping symbol differs from one where there is no grouping symbol. 4x3 differs from (4x)3because the exponent with a grouping symbol raises each factor to that power. In this case (4x)3 = 43 x3 = 64x3 KEEP in MIND that any variable without an exponent is assumed to be 1. In this example, (4x)3, the factors inside the parentheses have an exponent of 1. As a result, we have (41x1)3which gives us the value as shown above 42 ● 13 + 8 4 ● 4 ● 1 ● 1 ● 1 + 8 16 ● 1 + 8 = 24

  8. Common Assumptions with Numbers + 1 n. 1 1 • The sign of a number is positive, + • The coefficient is 1 • The decimal point is to the right of the number • As a whole number it is over 1 • The power of the number is 1

  9. 1.1 Real Numbers and Number Operations Whole numbers = 0, 1, 2, 3 … Integers = …, -3, -2, -1, 0, 1, 2, 3 … Rational numbers = numbers such as 3/4 , 1/3, -4/1 that can be written as a ratio of the two integers. When written as decimals, rational numbers terminate or repeat, 3/4 = 0.75, 1/3 = 0.333… Irrational numbers = real numbers that are NOT rational, such as, and  , When written as decimals, irrational numbers neither terminate or repeat. A Graph of a number is a point on a number line that corresponds to a real number The number that corresponds to a point on a number line is the Coordinate of the point. Origin  -3 -2 -1 0 1 2 3

  10. 1.1 Real Numbers and Number Operations Graph- 4/3, 2.7, • • • Graph- 2, 3 • • Graph- 1, - 3 • •

  11. 1.1 Real Numbers and Order of Operation Example: You can use a number line to graph and order real numbers. Increasing order (left to right): - 4, - 1, 0.3, 2.7 Properties of real numbers include the closure, commutative, associative, identity, inverse and distributive properties.

  12. 1.1 Using Properties of Real Numbers Properties of addition and multiplication [let a, b, c = real numbers] Property Addition Multiplication Closure a + b is a real number a • b is a real number Commutative a + b = b + a a• b = b • a Associative ( a + b ) + c = a + ( b + c ) ( a b ) c = a ( b c ) Identity a + 0 = a , 0 + a = a a• 1 = a , 1 • a = a Inverse a + ( -a ) = 0 a • 1/a = 1 , a  0 Distributive a ( b + c) = a b + a c Opposite = additive inverse, for example aand- a Reciprocal= multiplicative inverse (of any non-zero #) for example a and 1/a Definition of subtraction:a – b = a + ( - b ) Definition of division:a / b = a 1 / b , b  0

  13. 1.1 Real Numbers and Number Operations Identifying properties of real numbers & number operations ( 3 + 9 ) + 8 = 3 + ( 9 + 8 ) 14 • 1 = 14 [ Associative property of addition ] [Identity property of multiplication ] Operations with real numbers: Difference of 7 and – 10 ? 7 – ( - 10 ) = 7 + 10 = 17 • • Quotient of - 24 and 1/3 ?

  14. 1.1 Real Numbers and Number Operations • Give the answer with the appropriate unit of measure • A.) 345 miles – 187 miles=158 miles • B.) ( 1.5 hours ) ( 50 miles ) =75 miles • 1 hour • 24 dollars =8 dollars per hour • 3 hours • D) ( 88 feet ) ( 3600 seconds ) ( 1 mile ) =60 miles per hour • 1 second 1 hour 5280 feet “Per”means divided by

  15. 1.1 Solve Linear Equations Identifying Properties • – 8 + 8 = 0 • ( 3 • 5 ) • 10 = 3 • ( 5 • 10 ) • 7 • 9 = 9 • 7 • ( 9 + 2 ) + 4 = 9 + ( 2 + 4 ) • 12 (1) = 12 • 2 ( 5 + 11 ) = 2 • 5 + 2 • 11

  16. 1.1 Solve Word Problems Operations 43. What is the sum of 32 and – 7 ? 44. What is the sum of – 9 and – 6 ? 45. What is the difference of – 5 and 8 ? 46. What is the difference of – 1 and – 10 ? 47. What is the product of 9 and – 4 ? 48. What is the product of – 7 and – 3 ? 49. What is the quotient of – 5 and – ½ ? 50. What is the quotient of – 14 and 7/4 ?

  17. 1.1 Solve Unit Measures Unit Analysis • 8 1/6 feet + 4 5/6 feet = • 27 ½ liters – 18 5/8 liters = • 8.75 yards ( $ 70 ) = • 1 yard • ( 50 feet ) ( 1 mile ) ( 3600 seconds ) = • 1 second 5280 feet 1 hour

  18. 1.2 Algebraic Expressions and Models • Order of Operations • First, do operations that occur withingrouping symbols - 4 + 2 ( -2 + 5 ) 2 = - 4 + 2 (3 ) 2 • Next, evaluatepowers = - 4 + 2 ( 9 ) • Domultiplicationsanddivisionsfrom left to right = - 4 +18 • Doadditionsandsubtractions from left to right= 14 • Numerical expression: 25= 2 • 2 • 2 • 2 • 2 • [ 5 factors of 2 ] or [ 2 multiplied out 5 times ] • In this expression: • the number2is the base • the number5is theexponent • the expression is apower. • A variable is a letter used to represent one or more numbers. Any number used to replace variable is a value of the variable. An expression involving variables is called an algebraic expression. The value of the expression is the result when you evaluate the expression by replacing the variables with numbers. • An expression that represents a real-life situation is a mathematical model. See page 12.

  19. 1.2 Algebraic Expressions and Models Example: You can use order of operations to evaluate expressions. Numerical expressions: 8 (3 + 42) – 12  2 = 8 (3 + 16) – 6 = 8 (19) – 6 = 152 – 6 = 146 Algebraic expression: 3 x2 – 1 when x = – 5 3 (– 5 )2 – 1 = 3 (25) – 1 = 74 Sometimes you can use the distributive property to simplify an expression. Combine like terms: 2 x2 – 4 x + 10 x – 1 = 2 x2 + (– 4 + 10 ) x – 1 = 2 x2 + 6 x - 1

  20. 1.2 Evaluating Powers • Example 1: ( - 3 ) 4 = ( - 3 ) ( - 3 ) ( - 3 ) ( - 3 ) = 81 • - 3 4 = - ( 3 3 3 3 ) = - 81 • Example 2: Evaluating an algebraic expression • - 3 x2 – 5 x + 7 when x = - 2 • - 3 ( - 2 ) 2 – 5 ( -2 )x + 7 [ substitute – 2 for x ] • - 3 ( 4 ) – 5 ( -2 )x + 7 [ evaluate the power, 2 2 ] • - 12 + 10 + 7 [ multiply ] • + 5 [ add ] • Example 3: Simplifying by combining like terms • 7 x + 4 x = ( 7 + 4 ) x [ distributive ] • = 11 x [ add coefficients ] • 3 n 2 + n – n 2 = ( 3 n 2 – n 2 ) + n [ group like terms ] • = 2 n 2 + n [ combine like terms ] • 2 ( x + 1 ) – 3 ( x – 4 ) = 2 x + 2 – 3 x + 12 [ distributive ] • = ( 2 x – 3 x ) + ( 2 + 12 ) [ group like terms ] • = - x + 14 [ combine like terms ]

  21. 1.3 Solving Linear Equations Transformations that produce equivalent equations Additional property of equality Add same number to both sides if a = b, then a + c = b + c Subtraction property of equality Subtract same number to both sides if a = b, then a - c = b - c Multiplication property of equalityMultiply both sides by the same number if a = b and c ǂ 0, then a • c = b • c Division property of equality Divide both sides by the same number if a = b and c ǂ 0, then a ÷ c = b ÷ c Linear Equations in one variable in form a x = b, where a & b are constants and a ǂ 0. A number is a solution of an equation if the expression is true when the number is substituted. Two equations are equivalent if they have the same solution.

  22. 1.3 Solve Linear Equations Solving for variable on one side [by isolating the variable on one side of equation] Example 1: 3 x + 9 = 15 7 3 x + 9 - 9 = 15 - 9 7 [ subtract 9 from both sides to eliminate the other term] 3 x = 6 7 7• 3 x = 7• 6 3 7 [ multiply both sides by 7/3, the reciprocal of 3/7, to get xby itself] x = 14 Example 2: 5 n + 11 = 7 n – 9 - 5 n - 5 n[ subtract 5 n from both sides to get the variable on one side ] 11 = 2 n – 9 + 9 + 9 [ add 9 to both sides to get rid of the other term with the variable] 20 = 2 n 2 2 [ divide both sides by 2to get the variablenby itself on one side] 10 = n

  23. 1.3 Solve Linear Equations Example: You can use properties of real numbers and transformations that produce equivalent equations to solve linear equations. Solve

  24. 1.3 Solve Linear Equations Equations with fractions Example 3: 1 x + 1 = x – 1 3 4 6

  25. 1.4 ReWriting Equations and Formulas Example: You can an equation that has more than one variable, such as a formula, for one of its variables. Solve the equation for y: 2 x – 3 y = 6 – 3 y = – 2 x + 6 y = 2 x – 2 3 Solve for the formula for the area of a trapezoid for h: A = 1 ( b1 + b2) h 2 2 A = ( b1 + b2) h 2 A = h ( b1 + b2)

  26. 1.4 ReWriting an Equation with more than 1 variable Solve : 7 x – 3 y = 8 for the variable y. 7 x – 3 y = 8 - 7 x - 7 x [ subtract 7 x from both sides to get rid of the other term] – 3 y = 8 – 7 x – 3 – 3 – 3 [divide both sides by – 3to get the variable x by itself on one side ] y = – 8 + 7 x 3 3 Calculating the value of a variable Solve: x + x y = 1 when x = – 1 and x = 3 x + x y = 1 [ first solve for yso that when you replace x with – 1 and 3, you also solve for y ] - x - x [ subtract x from both sides to get rid of the other term without yin it] x y = 1 – x x x [divide by xto get yby itself ] y = 1 – x when x = - 1, then y = - 2 and when x = 3, then y = - 2/3 x

  27. 1.4 Common Formulas Distance D = r td = distance, r = rate, t = time Simple interest I = p r tI = interest, p = principal, r = rate, t = time Temperature F = 9/5 C + 32F = degrees Fahrenheit, C = degrees Celsius Area of a Triangle A = ½ b hA = area, b = base, h = height Area of a Rectangle A = l wA = area, l = length, w = width Perimeter of Rectangle P = 2 l + 2 wP = perimeter, l = length, w = width Area of Trapezoid A = ½ ( b1 + b2 ) hA = area, b1 = 1 base, b2 = 2 base, h = height Area of Circle A = πr2A = area, r = radius Circumference of Circle C = 2 π rC = circumference, r = radius

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