12/8/2011. Semester Review. Linear Equations. Slope of a Line. rise run. m = = Slope of a line:

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1 Semester Review Fall 011 Linear Equations How can you tell a linear equation from other equations? 1. A linear equation can be written in the forms: GENERAL FORM: Ax + By = C where A, B, and C are constants and x and y are variables. (numbers) SLOPE-INTERCEPT FORM:. A linear equation graphs a straight line. y = mx + b where m is the slope and b is the y-intercept. POINT-SLOPE FORM: y y = m( x x ) 1 1 where m is the slope and (x 1,y 1 ) is a point. Slope of a Line Slope of a line: o ( x, y ) 1 1 y y1 m = = x x 1 rise run rise run o ( x, y ) Note: The slope of a line is the SAME everywhere on the line!!! You may use any two points on the line to find the slope. 1

2 Calculating Slope Example 1: Calculate the slope between the given points. (, 7) and (1, 7)? 4 Intercepts of a Line x-intercept: where the graph crosses the x- axis. The coordinates are (a, 0). y-intercept: where the graph crosses the y- axis. The coordinates are (0, b). 5 Recall that there are three possibilities for the manner in which the graphs of two linear equations could meet. The lines could intersect once not intersect at all (we have seen this) (be parallel) intersect an infinite number of times (the two lines are the same) consistent independent inconsistent independent consistent dependent consistent system a system of equation with a solution inconsistent system a system of equation without a solution dependent equations equations that are equivalent independent equations equations that are not equivalent Solving Systems of Linear Equations

3 Example: Use the elimination method to solve the system of linear equations. 4x + 6y = 5 8x = 9y Is the system consistent or inconsistent? Are the equations dependent or independent? 7. Solving Systems of Linear Equations Using the Elimination Method Solve With Calc. and Check Is it Bound/unbound?; Consistent/inconsistent and explain? Dependent/independent and explain?; What is the solution? Systems of Linear Inequalities that is unbound. Systems of Linear Inequalities Systems of Linear Inequalities that is bound.

4 Solve by Graphing and Check Explain reasoning Example (find the domain) Example : Find the domain of function f defined by Solution to Example x x = 0 (x )(x + 1) = 0 x = or x = 1 Know how to factor!!! 4

5 Example (find the domain) The domain is all values that x can take on. I cannot have a negative inside the square root. So I'll set the insides greater-than-or-equal-to zero, and solve. The result will be my domain: x + > 0 x > x < x < / Then the domain is all x < /. Examples: Simplify each of the following; express all answers so that exponents are positive. Whenever an exponent is 0 or negative, we assume the base is not zero. r r r r 4 7 r = 4 r 7 = 9 r 6xy 1 x y 6xxy = = y ( x y ) 4. Zero and Negative Integer Exponents Adding Fractions Add the two new numerators. Keep the new denominator. Do Not Add the Denominators! 5

6 Example: Graph each set of numbers on the number line. { π, - } 1 8,.75, 17, , Graph Real Numbers on the Number Line Algebraic expression - Translating from Verbal Form to Algebraic Form Addition Subtraction Multiplication Division sum difference product quotient plus minus times divided by added to less than twice, etc. ratio more than decreased by double, triple, etc. split k equal ways greater than reduced by of average (add first) increased by subtracted from increased by a exceeds less factor of decreased by a factor of 1.7 Solving Word Problems Involving Real Numbers An all-day parking meter takes only dimes and quarters. If it contains 100 coins with a total value of $14.50, how many of each type of coin are in the meter? Let x = number of quarters in the meter. Then 100 x= number of dimes in the meter. Now, 0.5x (100 x) = or 0.5x x = x = 4.50 x =4.50/0.15 = 0 Thus, there will be 0 quarters and 70 dimes 18 6

7 To solve a word problem involving percents, we must first change the percents to their decimal equivalents. Example: Give the decimal equivalent of 5%. 5 5% = 100 Percent means per 100 = = 0.05 To change a percent to its decimal equivalent, move the decimal point two places to the left. Examples: Change each percent to a decimal. 5% 50%.5% % 5. Change Decimals to Percents and Visa Versa To solve problems involving percents, we will use the formula: amount = percent base Examples: Solve each equation. Usually comes after the word of don t forget to change to a decimal What number is 60% of 00? What percent of 5 is 14?.6(00) = 10 5x = 14. Solve Word Problems that Involve Percents We will use the following formulas from business: interest = principal rate time OR i = prt profit = revenue - cost OR p = r c simple interest formula Don t forget to change your percent to a decimal before using it in a formula! retail price = cost + markup OR r = c + m We will use the following formulas from science: distance = rate time OR d = rt 5( F ) C = 9 Where C and F are the temperature in degrees Celsius and Fahrenheit respectively..5 Use Formulas to Solve Word Problems 7

8 Geometry Formulas to Know Rectangle Perimeter = sum of all sides or Perimeter = l + w in units Rectangular Box Volume = l w h in units Area = l w in units.5 Use Formulas to Solve Word Problems Geometry Formulas to Know Rectangle Perimeter = sum of all sides or Perimeter = l + w in units Rectangular Box Volume = l w h in units Area = l w in units.5 Use Formulas to Solve Word Problems Geometry Formulas to Know Triangle Perimeter = sum of all sides Parallelogram Perimeter = sum of all sides in units in units 1 Area = bh Area = bh.5 Use Formulas to Solve Word Problems For these formulas: The height meets the base-side at a 90-degree angle 8

9 Geometry Formulas to Know Trapezoid Perimeter = sum of all sides a+b Area = h Circle Circumference Area = π r = π d in units π π 7 *To minimize roundingerrors we will use the π button on our TI.5 Use Formulas to Solve Word Problems Geometry Formulas to Know base * Pyramid Volume = hb 1 B is the area of the base * Cone Volume = 1 π r h.5 Use Formulas to Solve Word Problems Geometry Formulas to Know Sphere Volume = π r 4 Cylinder Volume = π r h.5 Use Formulas to Solve Word Problems 9

10 Example Zack is building a gate. It is to be five feet tall and eight feet wide. If the gate is square (that is, if the sides meet at the corners to form right angles), what will be the length of the diagonal bracing wire? Round to the nearest quarter-inch = c = 89 = c A collection of coins, consisting of nickels, dimes, and quarters, has a value of $.0. If there are three times as many nickels as quarters, and one-half as many dimes as nickels, how many coins of each kind are there? Step 1: number of quarters: q number of nickels: q number of dimes: (½)(q) = (/)q There is a total of coins, so: q + q + (/)q = 4q + (/)q = 8q + q = 66 11q = 66 q = 6 Then there are six quarters, and I can work backwards to figure out that there are 9 dimes and 18 nickels. How many ounces of pure water must be added to 50 ounces of a 15% saline solution to make a saline solution that is 10% salt? ounces liquid % salt total ounces salt water x % sol'n (50)(0.15) = % mix 50 + x (50 + x) From the last column, you get the equation 7.5 = 0.1(50 + x). Solve for x. 10

11 Helpful Hint for Factoring When factoring the sum or difference of two cubes, the sign between the terms in the binomial factor will be the same as the sign between the terms. The sign of the ab term will be the opposite of the sign between the terms of the binomial factor. The last term in the trinomial will always be positive. a + b = (a + b) (a ab + b ) same sign opposite sign always positive a b = (a b) (a + ab + b ) same sign opposite sign always positive Example X + 7x + 1 (x + )(x + ) Factors of 1 Sum of factors 1, 1 1, 6 8, 4 7 Factoring Trinomials x + x + x + Factored form (x + )(x + 1) = x + 4x + Notice the product of and 1 = and the sum of and 1 is 4 11

12 Product Rule for Radicals For nonnegative real numbers a and n a n b = n ab b, Examples: = 8 4 = 4 50 = 15 = = 16 = Product Rule for Radicals Examples: x y = 8x y = xy x y = 16 x y x y = x y x y Quotient Rule for Radicals Examples: 64 x 1 y 6 = 64 x 6 y 1 4 x = 4 y 5 64 x = x 64 x x 5 = x 1 = 16 x 1 = 4x 1

13 The Distance Formula d = (x x 1 ) + (y y 1 ) Example: Find the distance between the points (-4, 5) and (, -1) d = [ ( 4)] + ( 1 5) ~ Rules of Exponents For all real numbers a and b and all rational numbers m and n, Zero exponent rule: a 0 = 1, a 0 Raising a power to a power: Raising a product to a power : Raising a quotient to a power : m n m n ( a ) = a m m m ( ab ) = a b a b m a = b m m, b 0 The Product Rule for Exponents Multiplying exponents with the same base: x x 4 = x x x x x x x therefore, = x x x x x x x =? x 7 The product rule for exponents states: to multiply two exponential expressions with the same base, keep the common base and add the exponents. x m x n = x m+n 1

14 The Quotient Rule for Exponents State that when you divide like bases you subtract their exponents. If m and n represent natural numbers, m>n, and x 0, then x x m n = x m n NOTE: you always take the numerator s exponent minus your denominator s exponent, NOT the other way around. The Power Rule for Exponents States when you raise a base to two exponents, you multiply those exponents together. If m and n represent natural numbers, then m mn ( x ) n = x Example: 4 ( z ) x 8 Laws of Exponents: For any integers m, n (assuming no divisions by 0) Laws of Exponents m n x x = x x m n = m ( x ) n = ( ) n xy = Add Subtract Multiply Multiply examples n x = y Multiply 14

15 Polynomial Degrees Second-degree polynomial, 4x, x 9, or ax + bx + c Third-degree polynomial, 6x or x 7 Fourth-degree polynomial, x 4 or x 4 x + 9 Fifth-degree polynomial, x 5 or x 5 4x x + 7 Monomial, Binomial, and Trinomial Type Definition Example Monomial A polynomial with one term 5x Binomial A polynomial with two terms 5x 10 Trinomial A polynomial with three terms Examples h( x) = x + x x + Find: a) h(0) b) h(-) 15

16 Special Products Square of the sums: (x + y) = x + xy +y The square of the differences: (x y ) = x xy +y Product of the sum and difference of two terms: (a + b)(a b) = a b Identifying Polynomials The degree of a term of a polynomial in one variable is the exponent on the variable in that term. Example: 5x 6 (Sixth) 4x (Third) 7x (First) 9 (Zero) The degree of a polynomial is the same as that of its highest-degree term. Example: 5x 6 + 4x 7x + 9 (Sixth) Dividing Polynomials To divide a polynomial by a polynomial, use the same method as when performing long division. Divide. 6t - t - 40 t + 5 t = t + 5 6t - t t + 15t -16t - 40 dividend divisor 1. Divide 6t by t. Write the quotient above the term containing the t.. Multiply the t by t + 5. Write the product under the like terms.. Subtract. Bring down the remaining term. 16

17 Dividing Polynomials Divide. 6t - t - 40 t + 5 t - 8 = t + 5 6t - t t +15t -16t Repeat, using the first term in the bottom row: -16t t t = t - t - 40 t + 5 = t 8 Check your answer using the FOIL method. "Percent of" Word Problems What percent of 0 is 0? 0 = (x)(0) 0 0 = x = 1.5 So Thirty is 150% of 0 What is 5% of 80? 45% of what is 9? x = (0.5)(80) x = 8 Twenty-eight is 5% of = (0.45)(x) = x = 0 Nine is 45% of 0. "Percent of" Word Problems A computer software retailer used a markup rate of 40%. Find the selling price of a computer game that cost the retailer $5. The markup is 40% of the $5 cost, so the markup is: (0.40)(5) = 10 Then the selling price, being the cost plus markup, is: = 5 The item sold for $5. 17

18 Word Problem: Age In three more years, Jon's grandfather will be six times as old as Jon was last year. When Jon's present age is added to his grandfather's present age, the total is 68. How old is each one now? One-half of Heather's age two years from now plus onethird of her age three years ago is twenty years. How old is she now? Word Problem: Area A square has an area of sixteen square centimeters. What is the length of each of its sides? Word Problem: Coin A collection of coins, consisting of nickels, dimes, and quarters, has a value of $.0. If there are three times as many nickels as quarters, and one-half as many dimes as nickels, how many coins of each kind are there? A wallet contains the same number of pennies, nickels, and dimes. The coins total $1.44. How many of each type of coin does the wallet contain? 18

19 Word Problem: Distance A 555-mile, 5-hour plane trip was flown at two speeds. For the first part of the trip, the average speed was 105 mph. Then the tailwind picked up, and the remainder of the trip was flown at an average speed of 115 mph. For how long did the plane fly at each speed? Word Problem: Distance Two cyclists start at the same time from opposite ends of a course that is 45 miles long. One cyclist is riding at 14 mph and the second cyclist is riding at 16 mph. How long after they begin will they meet? Word Problem: Mixture How many liters of a 70% alcohol solution must be added to 50 liters of a 40% alcohol solution to produce a 50% alcohol solution? How many ounces of pure water must be added to 50 ounces of a 15% saline solution to make a saline solution that is 10% salt? 19

20 Word Problem: Numbers The product of two consecutive negative even integers is 4. Find the numbers. The sum of two consecutive integers is 15. Find the numbers. Word Problem: Investment You put $1000 into an investment yielding 6% annual interest; you left the money in for two years. How much interest do you get at the end of those two years? You have $50,000 to invest, and two funds that you d like to invest in. The You-Risk-It Fund (Fund Y) yields 14% interest. The Extra-Dull Fund (Fund X) yields 6% interest. Because of tax implications, you don't think you can afford to earn more than $4,500 in interest income this year. How much should you put in each fund? Solving Linear Equations Solve 7x + = 54 Solve 5 + 4x 7 = 4x x 0

21 Graph Graph 4x y = 1 Graph x = 4 Graph y = 0 Slope-Intercept Form Find the equation of the straight line that has slope m = 4 and passes through the point ( 1, 6). Slope-Intercept Form Find the equation of the line that passes through the points (, 4) and (1, ). 1

22 Solving Linear Inequalities Solve x + < 0, give interval notation, and graph Solving Linear Inequalities Solve 10 < x + 4 < 19 Things to remember 1. Show work and check. Interval notation (x, y). Use ruler, bring extra batteries 4. Pencils

23 Preparing for the test Review previous homework Review class notes Review concepts and definitions Complete the Chapter Review Place yourself in test-like conditions Get a good night s sleep Allow plenty of time to arrive for test Taking the test Read directions carefully Read each problem carefully Watch your time If you have time, check work and answers Do not turn your test in early If possible, double-check your work