Equations

I.E.S. Izpisúa Belmonte

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Matemáticas 3º ESO sección europea

Linear equations.

Solving linear equations.

An equation is a mathematical statement that has an expression on the

left side of the equals sign (=) with the same value as the expression on the

right side.

One of the terms in an equation may not be known and needs to be

determined. Often this unknown term is represented by a letter such as "x".

(e.g. x+5=2)

Solving an equation means manipulating the expressions and finding the

value of the unknown variables. In order to solve for the unknown variable,

you must isolate the variable

To keep an equation equal, we must do exactly the same thing to each side

of the equation. If we add (or subtract) a quantity from one side, we must

add (or subtract) that same quantity from the other side.

Equations with parentheses. Steps to solve an equation with

parentheses:

1. Rewrite the equation without parentheses by using the distributive

property.

2. Combine like terms.

3. Collect variables on one side and constants on the other.

4. Divide by the coefficient of the variable.

Equations with fractions. Steps to solve an equation with fractions:

1. Rewrite without fractions by multiplying both sides by the least common

denominator.

2. Combine like terms.

3. Collect variables on one side and constants on the other.

4. Divide by the coefficient of the variable.

Exercises.

1.- Add the same number to 100 and to 20 so as to make the greater number three times the smaller one. (Diofanto, 3rd century).

2.-The sum of three consecutive numbers is 87. What are they?

3.- The sum of three consecutive numbers is 72.What are the smallest of these numbers?


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4.- Jane spent 42€ on shoes. This was 14€ less than twice what she spent

on a blouse. How much was the blouse?

5.- The sum of two numbers is 84, and one of them is 12 more than the

other. What are the two numbers?

6.- The sum of two consecutive numbers is 37. What are they?

7.- Divide 80€ among three people so that the second will have twice as

much as the first, and the third will have 5€ less than the second.

8.- Julie has $50, which is eight dollars more than twice what John has.

How much has John?

9.- Carlotta spent $35 at the market. This was seven dollars less than three times what she spent at the bookstore, how much did she spend there?

10.- Jane spent $100 on books. This was 20 dollars less than five times what she spent on lunch. How much did she spend on lunch?

11.- The sum of two numbers is 99, and one of them is 17 more than the other. What are the two numbers?

12.- A class of 50 students is divided into two groups; one group has eight

less than the other. How many are in each group?

13.- The sum of two numbers is 72, and one of them is five times the other.

What are the two numbers?

14.- A group of 266 persons consists of men, women and children. There are four times as many men as children, and twice as many women as

children. How many of each are there?

15.- Half of a number, added to a fifth of three less than the number, is

equal to two thirds the number. What is the number?

16.-During an amorous struggle, the lady’s pearls broke. Half of the pearls fell onto the floor, a fourth rolled under a chair, a sixth fell into her lap, and

three pearls remained on the strand. How many pearls were there originally on the strand?

17.- Half of a number added to its third part, is eight less than the number. What is the number?

18.-Three fifths of a number plus 8 is equal to the number. What is the

number?

19.-Half of a number, plus a fifth of two less than the number, is four less

than the number. What is the number?

20.- Melissa went shopping and spent half of her money on shoes, a third on a blouse, a tenth to take her boyfriend to lunch, and she came home

with €12. How much did she start out with?

21.- Find two consecutive integers whose sum is 45.

22.- Find three consecutive even(even means par) integers whose sum is 72.

23.- Find two consecutive odd (odd means impar) integers whose sum is –88.


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24.-Find four consecutive odd integers whose sum is 56.

25.-Find two consecutive even integers such that the sum of the larger and twice the smaller is 62.

26.-Find three consecutive integers such that the sum of twice the smallest and 3 times the largest is 126.

27.-Seven times a number is equal to 12 more than 3 times the number.

Find the number.

28.-Ten more than 6 times a number is 4 less than 4 times the number.

29.-The second of two numbers is 4 times the first. Their sum is 50. Find the numbers.

30.-The second of two numbers is 5 more than twice the first. Their sum is

80. Find the numbers.

31.-The perimeter of a rectangle is 24 inches. Find the dimensions if its

length is 3 inches greater than its width.

32.-Find the measures of the angles of a triangle if the angles are represented by x, 4x and 4x.

33.-The perimeter of a triangle is 51 centimeters. The lengths of its sides are consecutive odd integers. Find the lengths of all three sides.

34.-Two-thirds times a number plus 7 equals 7 minus the number. Find the number.

35.- If the same number is added to the numerator and denominator of 7/9

the result is 5/6. What is the number?

36.-How many litres of wine at 4 euros/ litre would be necessary to mix

with 60 litres of wine at 3 euros/litre to get a blend that would value 3.6 euros a litre?

37.-. How many kilograms of coffee at 15 euros/kg would be necessary to

mix with 80 kg of coffee at 12 euro/kg to get a blend that would value 13 euros/kg?

38.-A shop owner mixes 35 kg of one blend of coffee that costs 12.5 €/kg, with 15kg of another blend of coffee that costs 14.5 €/kg. How much will this new blend of coffee cost?

39.-In a winery, they mix 100 litres of one barrel of wine that costs 3.5 €/litre with50 litres of another barrel of wine that costs 5.6€/litre How much

will this new wine blend cost?

40.-In a perfume factory, they mix 15 litres of one cologne that costs 60€/litre with25 litres of a different cologne that costs 50€/litre. How much

will this new cologne blend cost?

41.- How many litres of a 70% alcohol solution must be added to 50 litres

of a 40%alcohol solution to produce a 50% alcohol solution?

42.-How many pounds of lima beans that cost $0.90 per pound must be

mixed with 16pounds of corn that costs $0.50 per pound to make a mixture of vegetables that costs$0.65 per pound?


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43.-Two hundred litres of a refreshment that contains 35% fruit juice is

mixed with 300 litres (L) of another refreshment. The resulting fruit refreshment is 20% fruit juice. Find the percent of fruit juice in the 300

litres of refreshment.

44.-Ten grams of sugar are added to a 40-g serving of a breakfast cereal that is 30% sugar. What is the percent concentration of sugar in the

resulting mixture?

45.-Suppose you work in a lab. You need a 15% acid solution for a certain

test, but you only have a 10% acid solution and a 30% solution. So, you decide to mix 10% solution with 30% solution, to make your own 15% acid solution. You need 10 litres of the 15% acid solution. How many litres of

10% solution and 30% solution should you use?

46.- Solve these equations:

a. 53

4x2

4

9x3

2

7x5

b. 4

32

3

3

2

xx

xx

c. 26

5

3

5

xx

d. 6

32

6

)1()2(3

3

)3(2

4

52 2

xxxxx

e. 2

375)3(2

4

12

xx

x

f. 12

1

6

)2(5

6

)2(

3

5 22

xxxx

g.

x2

8

1x3

6

2x

3

2

4

1x2


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VOCABULARY

-LINEAR EQUATION:ECUACIÓN LINEAL

-EQUALS SIGN:SIGNO DE IGUAL

-SOLVE AN EQUATION:RESOLVER UNA ECUACIÓN

-VALUE:VALOR

-UNKNOWN:DESCONOCIDO

-ISOLATE:AISLAR

-VARIABLE:VARIABLE

-PARENTHESES:PARÉNTESIS

-DISTRIBUTIVE PROPERTY:PROPIEDAD DISTRIBUTIVA

-LIKE TERMS:TÉRMINOS SIMILARES

-SPEND ON:GASTAR EN ALGO

-BOOKSTORE:LIBRERÍA

-AMOROUS STRUGGLE:PELEA DE ENAMORADOS

-STRAND:HILO(DE UN COLLAR)

-EVEN NUMBER:PAR

-ODD NUMBER:IMPAR

-BLEND:MEZCLA

-SHOP OWNER:DUEÑO DE UNA TIENDA

-WINERY:BODEGA(AM.E)

-A BARREL OF WINE:BARRIL DE VINO

-LIMA BEANS:JUDÍAS BLANCAS

Equations

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