Solving Word Problems Using Equations

Solving Word Problems Using Equations María Olivencia Manzano Paula Olivencia Peña Laura Rodríguez González Ángela Rodríguez Pomares 4ºD

INDEX INTRODUCTION -----------------------------------------------------------------

3

HOW TO SOLVE WORD PROBLEMS? -----------------------------------

4

EQUATIONS ----------------------------------------------------------------------

5

Linear equations ---------------------------------------------------------

5-8

Quadratic equations ----------------------------------------------------

9-11

SYSTEMS OF EQUATIONS --------------------------------------------------

12-13

Linear systems of equations -----------------------------------------

14-15

Nonlinear systems of equations -------------------------------------

16

INEQUALITIES -------------------------------------------------------------------

17

SOLUTIONS ----------------------------------------------------------------------

18

BIBLIOGRAPHY -----------------------------------------------------------------

19-20

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INTRODUCTION HELLO GUYS! We are going to talk about solving word problems using equations.

In our work, we include word problems about linear equations, quadratic equations, systems of equations, non linear systems of equations and linear inequalities.

If you don’t know how to do them, we have included some instructions to solve these word problems. We explain everything what we have named previously.

The problems are ordered from easier to more difficult.

We have tried to do this workbook fun for you. We hope it helps you if you don’t understand this matter.

At the end of the workbook, you can find the solutions of the problems pointed with the symbol:

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How to Solve Word Problems?

Step 1: Read the problem as many times as you need and try to, answer these questions: 1. What are we trying to find?

2. What is the problem telling me that is useful? (Cross out unneeded information).

3. If it is possible, draw a simple picture of the problem to make it more real to you.

Step 2: Assign variables and write down what the variables represent. Use as few unknowns as possible. If you can represent all the unknown information in terms of a single letter do so!

Step 3: Translate the word terms into algebraic equations. Remember, you have to create as many separate equations as you have unknowns.

Step 4: Solve the equation, system or whatever you go using the rules of algebra.

Step 5: Answer the question in the problem and check it to see if it makes common sense.

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EQUATIONS An equation is a mathematical statement that has two algebraic expressions separated by an equal sign.

The expression on the left side of the equal sign is the left-hand side of the equation, and it has the same value as the expression on the right side (righthand side).

The degree of an equation that has not more than one variable in each term is the exponent of the highest power to which that variable is raised in the equation.

LINEAR EQUATIONS o What is it?

It is an equation that makes a straight line when it is graphed.

Often written in the form: y = mx+b Example: y = 2x+1 is a linear equation.

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o Now, we are going to show you some examples of word problems using linear equations: 1. When 6 is added to four times a number, the result is 50. Find the number.

Step 1: Read the problem as many times as you need and try to, answer these questions: 1. What are we trying to find?

2. What is the problem telling me that is useful? (Cross out unneeded information).

3. If it is possible, draw a simple picture of the problem to make it more real to you.

Step 2: Assign variables and write down what the variables represent. Use as few unknowns as possible. If you can represent all the unknown information in terms of a single letter do so!

Step 3: Translate the word terms into algebraic equations. Remember, you have to create as many separate equations as you have unknowns.

Step 4: Solve the equation, system or whatever you go using the rules of algebra.

Step 5: Answer the question in the problem and check it to see if it makes common sense.

n = a number

The number we are looking for is 11.

Check the answer:

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2. The length of a rectangular map is 15 inches and the perimeter is 50 inches. Find the width.

Step 1: Read the problem as many times as you need and try to, answer these questions: 1. What are we trying to find?

2. What is the problem telling me that is useful? (Cross out unneeded information).

3. If it is possible, draw a simple picture of the problem to make it more real to you.

Step 2: Assign variables and write down what the variables represent. Use as few unknowns as possible. If you can represent all the unknown information in terms of a single letter do so!

Step 3: Translate the word terms into algebraic equations. Remember, you have to create as many separate equations as you have unknowns.

Step 4: Solve the equation, system or whatever you go using the rules of algebra.

Step 5: Answer the question in the problem and check it to see if it makes common sense.

w = the width of a rectangle Perimeter = width + length + width + length. (

)

The width of the rectangle is 10 inches.

Check the answer:

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o LET’S PRACTICE

✔1. John’s mom runs a dairy farm. Last year Betty the cow gave 375 gallons less than twice the amount from Bessie the cow. Together, Betty and Bessie produced 1464 gallons of milk. How many gallons did each cow give?

✔2. Twice a number is added to the number and the answer is 90. Find the number.

✔3.

Luke has a board that is 44 inches long. He wishes to cut it into two

pieces so that one piece will be 6 inches longer than the other. How long should the shorter piece be?

✔4.

On an algebra test, the highest grade was 42 points higher than the

lowest grade. The sum of the two grades was 138. Find the lowest grade.

✔5.

Kim and Cyndi are starting a business tutoring students in math. They

rent an office for $400 per month and charge $40 per hour per student. If they have 15 students each for one hour per week how much profit do they make together in a month? (assume 4 weeks per month)

✔6. Find three consecutive integers with a sum of 243. ✔7. Brady has $66 in his account. He is saving $4.20 per week. How long will it take him to save $339?

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QUADRATIC EQUATIONS o WHAT IS IT? It is an equation where the highest exponent of the variable (usually "x") is a square.

The quadratic formula is a general way of solving any quadratic equation:

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o Now, we are going to show you some examples of word problems using quadratic equations: 1. The width of a rectangle is 16 feet less than 3 times the length. If the area is 35 square feet, find the dimensions of the rectangle.

Step 1: Read the problem as many times as you need and try to, answer these questions: 1. What are we trying to find?

2. What is the problem telling me that is useful? (Cross out unneeded information).

3. If it is possible, draw a simple picture of the problem to make it more real to you.

Step 2: Assign variables and write down what the variables represent. Use as few unknowns as possible. If you can represent all the unknown information in terms of a single letter do so!

Step 3: Translate the word terms into algebraic equations. Remember, you have to create as many separate equations as you have unknowns.

Step 4: Solve the equation, system or whatever you go using the rules of algebra.

Step 5: Answer the question in the problem and check it to see if it makes common sense.

x = length

35 = x(3x -16) 35 = 3x2 - 16x

Only the second factor will give a positive solution, so the answer is 7. The dimensions of the rectangle are: Length: 7 feet/ Width: 5 feet.

0 = (3x +5)(x -7).

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o LET’S PRACTICE

✔1.

A garden measuring 12 meters by 16 meters is to have a pedestrian

pathway installed all around it, increasing the total area to 285 square meters.

What will be the width of the pathway?

✔2. You have to make a square-bottomed, box with a height of three inches and a volume of approximately 42 cubic inches. You will be taking a piece of cardboard, cutting three-inch squares from each corner, scoring between the corners, and folding up the edges. What should be the dimensions of the cardboard, to the nearest quarter inch?

✔3.

Andrew and Bill, working together, can cover the

roof of a house in 6 days. Andrew, working alone, can complete this job in 5 days less than Bill. How long will it take Bill to make this job?

✔4. Two tubes, working together, can fill the

reservoir with the liquid in 12

hours. The larger tube, if works separately, can fill the reservoir in 18 hours less than the smaller tube. How long will it take to fill the reservoir using the smaller tube only?

✔5. Find two positive consecutive odd integers whose product is 99. ✔6. A rectangular solar heat panel has a length 3 m more than its width. If the area of the solar panel is 28 m², how long is the panel?

✔7. The area of a rectangle is 560 square inches. The length is 3 more than twice the width. Find the length and the width.

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SYSTEMS OF EQUATIONS Two equations for which you need a common solution are called a pair of simultaneous equations or system of equations.

There is more than one way to solve any pair of simultaneous equations: -

The graphical method: to solve them approximately.

The graphical solution of linear simultaneous equations is the point of intersection found by drawing the two linear equations on the same axes.

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-

An algebraic method (by elimination or by substitution): to solve them exactly.

Substitution method for solving systems of equations: You substitute one of the unknowns with an equivalent expression or value.

Elimination method (or addition method) for solving systems of equations: The original equations are combined to eliminate one of the unknowns making an equation that is easier to solve.

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o

Now, we are going to show you some examples of word problems using systems of equations:

1. If 4 apples and 2 oranges equals $1 and 2 apples and 3 orange equals $0.70, how much does each apple and each orange cost?

Step 1: Read the problem as many times as you need and try to, answer these questions: 1. What are we trying to find?

2. What is the problem telling me that is useful? (Cross out unneeded information).

3. If it is possible, draw a simple picture of the problem to make it more real to you.

Step 2: Assign variables and write down what the variables represent. Use as few unknowns as possible. If you can represent all the unknown information in terms of a single letter do so!

Step 3: Translate the word terms into algebraic equations. Remember, you have to create as many separate equations as you have unknowns.

Step 4: Solve the equation, system or whatever you go using the rules of algebra.

Step 5: Answer the question in the problem and check it to see if it makes common sense. x= price for one apple y= price for one orange

*Apply the elimination method.

Calculate x :

Each apple costs 20 cents and each orange costs 10 cents.

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o LET’S PRACTICE 1. A company employs 60 people. Of this amount, 16% of the men wear glasses and 20% of the women also wear glasses. If the total number of people who wear glasses is 11, how many men and women are there in the company?

2. Two appliances have been purchased for $3,500. If a 10% discount was applied to the first item and a 8% discount on the second, the total price for both purchases would have been $3,170. What is the price of each item?

3. Three types of grain are sold by a farmer: wheat, barley and millet. Each portion of wheat sells for $4.00, the barley for $2.00 and the millet for $0.50. If he sells 100 portions in total and receives $100 from the sale, how many portions are sold of each type?

4. A collection of 105 coins consists of 1.00 dollar and 5.00 dollars. If the total value is 205.00 dollars, find the number of coins of each denomination in the collection.

5. A vending machine takes only nickels and dimes. There are 5 times as many dimes as nickels in the machine. The face value of the coins is $4.40.How many of each coin are in the machine.

6. Two angles are complementary, one angle is 42 degrees more than half the other. Find the angles.

7. A two digit number is 6 times the sum of its digits. The tens digit is 1 more than the units digit. Find the original number.

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NONLINEAR SYSTEMS OF EQUATIONS An equation in which one or more terms have a variable of degree 2 or higher is called a non linear equation.

A nonlinear system of equations contains at least one nonlinear equation.

WHEN SOLVING A SYSTEM IN WHICH ONE EQUATION IS LINEAR, IT IS EASIEST TO USE THE SUBSTITUTION METHOD.

o LET’S PRACTICE 1. The product of two numbers is 4, and the sum of their squares is 17.

What are these numbers?

2. Find an equivalent fraction to 5/7 whose terms raised to the power of two added are 1184.

3. A group of students organize a field trip and they rent a bus for 540€.

On the day they had to leave, 6 students missed it and this made each one to pay 3€ more. Calculate the number of students who went to the trip and how much each one paid.

4. The area of a rectangular garden is 900 m² and it is surrounded by a walkside which is 5 m wide and its area is 850 m². Calculate the dimensions of the garden.

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INEQUALITIES An inequality is a mathematical statement that has two algebraic expressions separated by <, >, ≤, or ≥.

LINEAR INEQUALITIES IN ONE VARIABLE Solving linear inequalities is very similar to solving linear equations, except for one small but important detail: you flip the inequality sign whenever you multiply or divide the inequality by a negative.

o LET’S PRACTICE

✔1. Five more than twice a number is at least 45. What is the minimum value of the number?

2. There are dogs and cats in a kennel. The number of cats is twice the number of dogs. What is the greatest number of cats in the kennel if there are at most a total of 48 animals in the kennel?

✔3.

In a remedial math class at the community college, only five chapter

exams are given and an 80% must be achieved in order to take the state exit exam. Pam has completed the first four exams with scores of 71, 84, 79, and 81. Write an inequality to find the minimal score Pam can make on the fifth exam in order to take the state exit exam?

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SOLUTIONS  Linear Equations: Problem 1: Bessie: 613 gallons of milk, Betty: 851 gallons of milk.

Problem 2: 30 Problem 3: 19 inches Problem 4: 48 Problem 5: $2000 Problem 6: 65 weeks Problem 7: 80, 81 and 82

 Quadratic Equations: Problem 1: 1.5 meters Problem 2: 9.75 inches Problem 3: 15 days Problem 4: 36 hours Problem 5: 9 and 11 Problem 6: 7 m Problem 7: Width: 16 inches/ Length: 35 inches

 Inequalities: Problem 1: 20 Problem 3: 85%

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BIBLIOGRAPHY http://www.mathsisfun.com/ http://books.google.es/books?id=Zt_wbhEU6ewC&pg=PA60&lpg=PA60& dq=problemas+de+sistemas+de+ecuaciones+no+lineales+4o+eso&sou rce=bl&ots=Z27kF7-fRk&sig=hejwo-6xOEutZe0K0y-rPMp0PY&hl=en&sa=X&ei=HnMaUfXtK5Cr0AX34HoBg&ved=0CGIQ6AEwBg#v=onepage&q=problemas%20de%20siste mas%20de%20ecuaciones%20no%20lineales%204o%20eso&f=false http://www.juntadeandalucia.es/averroes/iesarroyo/matematicas/materi ales/4eso/solucionlibronuevo4b/U-3.pdf Paqui Cabrera’s notes

How to Solve Word Problems? http://academic.cuesta.edu/acasupp/as/706.htm http://www.studygs.net/mathproblems.htm

Linear Equations: http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_OneVariabl eWritingEquations.xml http://hotmath.com/help/gt/genericalg1/section_1_7.html http://www.algebralab.org/practice/practice.aspx?file=Word_LinearEquat ions.xml

Quadratic Equations: http://academic.cuesta.edu/smanor/math23dl/Quad/applications.htm http://www.regentsprep.org/Regents/math/ALGEBRA/AE5/PFacEq.htm http://www.purplemath.com/modules/quadprob2.htm http://www.algebra.com/algebra/homework/quadratic/lessons/Usingquadratic-equations-to-solve-word-problems.lesson http://www.millersville.edu/~bikenaga/basic-algebra/quadratic-wordproblems/quadratic-word-problems.html http://pg-forum.com/index.php?showtopic=2467 http://www1.broward.edu/~hsorkin/IntWeb/Quadratic_Word_Problems.htm

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Systems of Equations: http://www.vitutor.com/alg/system/problems_systems.html http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Systems %20of%20Equations%20Word%20Problems.pdf http://www.algebra.com/algebra/homework/coordinate/lessons/Solvingword-probs-using-linear-systems-of-two-eqns-with-two-unknowns.lesson

Nonlinear Systems of Equations: http://www.vitutor.com/alg/system/nolinear_systems.html

Inequalities: http://www.algebra-class.com/solving-word-problems-in-algebra.html http://www.education.com/study-help/article/inequality-wordproblems/?page=2 http://www.algebralab.org/practice/practice.aspx?file=word_inequalities. xml

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