UNIT 5. EQUATIONS AND SYSTEM OF EQUATIONS EQUATIONS An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign (=), for example asserts that x+3 is equal to 5. Knowns and unknowns Equations often express relationships between given quantities, the knowns, and quantities yet to be determined, the unknowns. By convention, unknowns are denoted by letters at the end of the alphabet, x, y, z, w,, while knowns are denoted by letters at the beginning, a, b, c, d,. The process of expressing the unknowns in terms of the knowns is called solving the equation. In an equation with a single unknown, a value of that unknown for which the equation is true is called a solution or root of the equation. Types of equations Equations can be classified according to the types of operations and quantities involved. Important types include: An algebraic equation is an equation involving only algebraic expressions in the unknowns. These are further classified by degree. A linear equation is algebraic equation of degree one. A polynomial equation is an equation in which a polynomial is set equal to another polynomial. A transcendental equation is an equation involving a transcendental function of one of its variables. A functional equation is an equation in which the unknowns are functions rather than simple quantities. A differential equation is an equation involving derivatives. An integral equation is an equation involving integrals. A Diophantine equation is an equation where the unknowns are required to be integers. A quadratic equation In this unit we are going to study the linear equation, the quadratic equation and some easy examples of polynomial equations. Exercise 1: Solve the following linear equations: 1) 3(x-2)+5(x-3) = 0 1
2) 3x + 2 = 5 3) 2x+4 = 3x + 1 4) 5-2(5x+1)+15x = 2 Exercise 2: Solve the following problems with equations 1) I think of a number, double it and add 7 and I have 3 times the number that I first thought of. What is the number? 2) Roland is three times Isabelle age now. 5 years ago, Roland was 4 times as old as Isabelle. How old are roland and Isabelle? 3) In 10 years time, Homere will be twice as old as he is now. How old is he now? 4) For which positive number is 16 times the cube root of the number the same as the number? Identities One use of equations is in mathematical identities, assertions that are true independent of the values of any variables contained within them. For example, for any given value of x it is true that Quadratic equation A quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the form where x represents a variable or an unknown, and a, b, and c are constants with a 0. (If a = 0, the equation is a linear equation.) The constants a, b, and c are called respectively, the quadratic coefficient, the linear coefficient and the constant term or free term. A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real. The roots are given by the quadratic formula where the symbol "±" indicates that both 2
are solutions of the quadratic equation. In the above formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using the Greek delta: A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases: If the discriminant is positive, then there are two distinct roots, both of which are real numbers: For quadratic equations with integer coefficients, if the discriminant is a perfect square, then the roots are rational numbers in other cases they may be quadratic irrationals. If the discriminant is zero, then there is exactly one distinct real root, sometimes called a double root: If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex roots Vieta's formulas give a simple relation between the roots of a polynomial and its coefficients. In the case of the quadratic polynomial, they take the following form: and These results follow immediately from the relation: 3
which can be compared term by term with: The first formula above yields a convenient expression when graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex, when there are two real roots the vertex s x-coordinate is located at the average of the roots (or intercepts). Thus the x-coordinate of the vertex is given by the expression: The y-coordinate can be obtained by substituting the above result into the given quadratic equation, giving SYSTEM OF EQUATIONS A system of equations is a collection of two or more equations with the same set of unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system. The equations in the system can be linear or non-linear. A system of linear equations can be solved four different ways: Substitution Elimination Matrices Graphing The Method of Substitution: The method of substitution involves several steps: Step 1: Solve for x in equation (1). 4
Step 2: Substitute this value for x in equation (2). This will change equation (2) to an equation with just one variable, y. Step 3: Solve for y in the translated equation (2). Step 4: Substitute this value of y in equation (1) and solve for x. Step 5: Check your answers by substituting the values of x and y in each of the original equations. If, after the substitution, the left side of the equation equals the right side of the equation, you know that your answers are correct. The Method of Elimination: In a two-variable problem rewrite the equations so that when the equations are added, one of the variables is eliminated, and then solve for the remaining variable. Step 1: Multiply equation (1) by -.09 and add it to equation (2) to obtain equation (3) with just one variable. 5
Step 2: Substitute in equation (1) and solve for x. Step 3: Check your answers. The method of Graphing: In this method solve for y in each equation and graph both. The point of intersection is the solution. Example: 1. Solve: y = x + 1 and y = -2x + 4 Solution: Begin by drawing a couple of tables (one for each equation) and filling them in. y = x + 1------------------ x 0 2-3 ------------------ y 1 3-2 ------------------ y = -2x + 4----------------------- x 0 2-2 4 ----------------------- y 4 0 8-4 ----------------------- Now, plot those points and draw a line connecting them. Once that has been done, you will see that the lines intersect at(1,2) (it is fairly obvious on the graph that it is exactly (1,2)). 6
Always be on the lookout for tricky situations, such as systems of equations that when graphed are two parallel lines. Since they're parallel, they will never intersect, and there will be no solution to that problem. Let s practice with Word Problems 1) The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many children and how many adults attended? 2) The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number. 3) Find the equation of the parabola that passes through the points ( 1, 9), (1, 5), and (2, 12). 4) A landscaping company placed two orders with a nursery. The first order was for 13 bushes and 4 trees, and totalled $487. The second order was for 6 bushes and 2 trees, and totalled $232. The bills do not list the per-item price. What were the costs of one bush and of one tree? 5) A passenger jet took three hours to fly 1800 miles in the direction of the jetstream. The return trip against the jetstream took four hours. What was the jet's speed in still air and the jetstream's speed? 6) A total of $12,000 is invested in two funds paying 9% and 11% simple interest. If the yearly interest is $1,180, how much of the $12,000 is invested at 9% and how much is invested at 11%? 7) An airplane flying into a headwind travels the 1800-mile flying distance between two cities in 3 hours and 36 minutes. On the return flight, the same distance is traveled in 3 hours. Find the ground speed of the plane and the speed of the wind, assuming that both remain constant. 8) Ten gallons of a 30% acid mixture is obtained by mixing a 20% solution with a 50% solution. How much of each must be used? 9) Five hundred tickets were sold for a certain music concert. The tickets for the adults and children sold for $7.50 and $4.00, respectively, and the total receipts for the performance were $3,312.50. How many of each kind of ticket were sold? Systems of Equations Word Problems 1) Find the value of two numbers if their sum is 12 and their difference is 4. 2) The difference of two numbers is 3. Their sum is 13. Find the numbers. 7
3) Flying to Kampala with a tailwind a plane averaged 158 km/h. On the return trip the plane only averaged 112 km/h while flying back into the same wind. Find the speed of the wind and the speed of the plane in still air. 4) The school that Stefan goes to is selling tickets to a choral performance. On the first day of ticket sales the school sold 3 senior citizen tickets and 1 child ticket for a total of $38. The school took in $52 on the second day by selling 3 senior citizen tickets and 2 child tickets. Find the price of a senior citizen ticket and the price of a child ticket. 5) The sum of the digits of a certain two-digit number is 7. Reversing its digits increases the number by 9. What is the number? 6) A boat traveled 210 miles downstream and back. The trip downstream took 10 hours. The trip back took 70 hours. What is the speed of the boat in still water? What is the speed of the current? 7) The state fair is a popular field trip destination. This year the senior class at High School A and the senior class at High School B both planned trips there. The senior class at High School A rented and filled 8 vans and 8 buses with 240 students. High School B rented and filled 4 vans and 1 bus with 54 students. Every van had the same number of students in it as did the buses. Find the number of students in each van and in each bus. 8) The senior classes at High School A and High School B planned separate trips to New York City. The senior class at High School A rented and filled 1 van and 6 buses with 372 students. High School B rented and filled 4 vans and 12 buses with 780 students. Each van and each bus carried the same number of students. How many students can a van carry? How many students can a bus carry? 9) Brenda's school is selling tickets to a spring musical. On the first day of ticket sales the school sold 3 senior citizen tickets and 9 child tickets for a total of $75. The school took in $67 on the second day by selling 8 senior citizen tickets and 5 child tickets. What is the price each of one senior citizen ticket and one child ticket? 10) Matt and Ming are selling fruit for a school fundraiser. Customers can buy small boxes of oranges and large boxes of oranges. Matt sold 3 small boxes of oranges and 14 large boxes of oranges for a total of$203. Ming sold 11 small boxes of oranges and 11 large boxes of oranges for a total of $220. Find the cost each of one small box of oranges and one large box of oranges. 11) A boat traveled 336 miles downstream and back. The trip downstream took 12 hours. The trip back took 14 hours. What is the speed of the boat in still water? What is the speed of the current? 8
12) DeShawn and Shayna are selling flower bulbs for a school fundraiser. Customers can buy bags of windflower bulbs and bags of daffodil bulbs. DeShawn sold 10 bags of windflower bulbs and 12 bags of daffodil bulbs for a total of $380. Shayna sold 6 bags of windflower bulbs and 8 bags of daffodil bulbs for a total of $244. What is the cost each of one bag of windflower bulbs and one bag of daffodil bulbs? 13) Two small pitchers and one large pitcher can hold 8 cups of water. One large pitcher minus one small pitcher constitutes 2 cups of water. How many cups of water can each pitcher hold?. 14) A test has twenty questions worth 100 points. The test consists of True/False questions worth 3 points each and multiple choice questions worth 11 points each. How many multiple choice questions are on the test?. 15) Margie is responsible for buying a week's supply of food and medication for the dogs and cats at a local shelter. The food and medication for each dog costs twice as much as those supplies for a cat. She needs to feed 164 cats and 24 dogs. Her budget is $4240. How much can Margie spend on each dog for food and medication? 16) Bill and Steve decide to spend the afternoon at an amusement park enjoying their favorite activities, the water slide and the gigantic Ferris wheel. Their tickets are stamped each time they slide or ride. At the end of the afternoon they have the following tickets: Fun Time Fun Time Amusements Amusements Water Slide: Water Slide: Ferris Wheel: Ferris Wheel: Total: $15.55 Total: $17.70 Steve's Ticket Bill's Ticket How much does it cost to ride the Ferris Wheel? How much does it cost to slide on the Water Slide? 17) The equations 5x + 2y = 48 and 3x + 2y = 32 represent the money collected from school concert tickets sales during two class periods. If x represents the cost for each adult ticket and y represents the cost for each student ticket, what is the cost for each adult ticket? 9
18) The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many children and how many adults attended? 19) The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number. 20) Find the equation of the parabola that passes through the points ( 1, 9), (1, 5), and (2, 12). 21) A landscaping company placed two orders with a nursery. The first order was for 13 bushes and 4 trees, and totalled $487. The second order was for 6 bushes and 2 trees, and totalled $232. The bills do not list the per-item price. What were the costs of one bush and of one tree? 22) A passenger jet took three hours to fly 1800 miles in the direction of the jetstream. The return trip against the jetstream took four hours. What was the jet's speed in still air and the jetstream's speed? 23) A total of $12,000 is invested in two funds paying 9% and 11% simple interest. If the yearly interest is $1,180, how much of the $12,000 is invested at 9% and how much is invested at 11%? 24) An airplane flying into a headwind travels the 1800-mile flying distance between two cities in 3 hours and 36 minutes. On the return flight, the same distance is traveled in 3 hours. Find the ground speed of the plane and the speed of the wind, assuming that both remain constant. 25) Ten gallons of a 30% acid mixture is obtained by mixing a 20% solution with a 50% solution. How much of each must be used? 26) Five hundred tickets were sold for a certain music concert. The tickets for the adults and children sold for $7.50 and $4.00, respectively, and the total receipts for the performance were $3,312.50. How many of each kind of ticket were sold? 10