unit 1 equation

8/7/2019 UNIT 1 Equation

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An equation is a proposition, expressing inalgebraic characters, the equality betweenone quantity or set of quantities and another,

or between different expressions for thesame quantity

An equation involving variables is called an

algebraic equation.


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Equations always have a LHS and RHS, else itwill only be an expression.

In the example x+ 3 = 7, x= 4 satisfies the

equation, 4 is known as the solution or rootof the equation

To solve an equations means to find the rootor solution of the equation.

The reduction of an equation consists, then,in bringing the unknown quantity by itself,on one side, and all the known quantities onthe other side, without destroying theequation.


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If the same or equal quantities be added toeach side

If the same or equal quantities be subtracted

from each side If each be multipliedby the same or equal

quantities side

If each be dividedby the same or equal

quantities side


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Ex. 1. Reduce the equation :

x + 3b - m = h - d

Transposing +3b, we have :

x - m = h - d - 3b

Transposing m we have:

x = h - d -3b + m.


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1. 3x + 6 = 5x + 7

2. 3x/4 + 6 = 5x/8 + 7

3. 40 - 6x - 16 = 120 - 14x

4. x/3 + x/5 = 20 - x/4

5. (1 - a)/x - 4 = 5

6. 6x/(x + 4) = 1

7. x + x/2 + x/3 = 118. (x - 5)/4 + 6x = (284 - x)/5

9. 3x + (2x + 6)/5 = 5 + (11x - 37)/2

10. (6x - 4)/3 - 2 = (18- 4x)/3 + x


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11. 3x - (x - 4)/4 - 4 = (5x + 14)/3 - 1/12

12. (7x + 5)/3 - (16 + 4x)/5 + 6 = (3x + 9)/2

13. x - (3x - 3)/5 + 4 = (20 - x)/2 - (6x - 8)/7 +

(4x - 4)/514. (6x + 7)/9 + (7x - 13)/(6x + 3) = (2x + 4)/3

15. 4x/5 -13x/6 +5 = 16x +6

16. 2x/11 + 3x /5 -5 =6x/2

17. 15x /2 -5= 4x/3 +5x/2 -618. 22x/3 -15x/2 -5 - 3x/2 =5x/2 -3

19. (a+b)x-(a-b)x =3x -5

20. (a+b) x + (a-b)x = (a+b)x +5


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Problem 1. A man being asked how much hegave for his watch, replied; If you multiply theprice by 4, and to the product add 70, and fromthis sum subtract 50, the remainder will beequal to 220 dollars.

Let the price of the watch be represented by xThis price is to be multiplied by 4, whichmakes 4xTo the product, 70 is to be added, making 4x +70

From this, 50 is to be subtracted, making 4x +70 - 50

We have, therefore, this equation 4x + 70 - 50= 220

Which reduced gives x = 50.


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Problem 2: What number is that, to which, ifits half be added, and from the sum 20 besubtracted, the remainder will be a fourth of

the number itself? In this problem, let x be put for the number

required.Then by the conditions proposed,

x + x/2 - 20 = x/4And reducing the equation x = 16.

Proof, 16 + 16/2 - 20 = 16/4.


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Problem 3. A father divides his estate among his threesons, in such a manner, that:The first has $1000 less than half of the whole;The second has 800 less than one third of the whole;The third has 600 less than one fourth of the whole;What is the value of the estate?

If the whole estate be represented by x, then theseveral shares will be x/2 - 1000, and x/3 - 800, and x/4- 600.

And as these constitute the whole estate, they aretogether equal to x.

We have then this equation x/2 - 1000 + x/3 - 800 + x/4 - 600 = x.

Which reduced gives x = 28800Proof 28800/2 - 1000 + 28800/3 - 800 + 28800/4 - 600 =28800.


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Problem 4: Divide 48 into two such parts,that if the less be divided by 4, and thegreater by 6, the sum of the quotients will

be 9Here, if x be put for the smaller part, the

greater will be 48 - x.

By the conditions of the problem

x/4 + (48 - x)/6 = 9.Therefore x = 12, the less.And 48 - x = 36, the greater.


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Problem 5. If to a certain number, 720 beadded, and the sum be divided by 125 ; thequotient will be equal to 7392 divided by462. What is that number?

Let x = the number required.a = 720 d = 7392b = 125 h = 462Then by the conditions of the problem

(x + a)/b = d/hTherefore x = (bd - ah)/hRestoring the numbers,

x = [(125.7392) - (720.462)]/462 = 1280.


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Problem 6. A merchant gains or loses, in abargain, a certain sum. In a second bargain,he gains 350 dollars, and, in a third, loses 60.

In the end he finds he has gained 200 dollars,by the three together. How much did he gainor lose by the first ?

If the profit is marked +, the loss must be -.

Let x = the sum required.Then according to the statement

x + 350 - 60 = 200And x = -90.


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Problem 7. A ship sails 4 degrees north, then13 S. then 17 N. then 19 S. and has finally 11degrees of south latitude. What was her

latitude at starting ? Let x = the latitude sought.Then marking the northings +,

and the southings -;

By the statement x + 4 - 13 + 17 - 19 = -11And x = 0.


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Problem 8. If a certain number is divided by12, the quotient, dividend, and divisor,added together, will amount to 64. What is

the number? Let x = the number sought.Then x/12 + x + 12 = 64.And x - 624/13 = 48.


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Problem 9. An estate is divided among fourchildren, in such a manner that,The first has 200 dollars more than 1/4 ofthe whole,The second has 340 dollars more than 1/5 ofthe whole,The third has 300 dollars more than 1/6 ofthe whole,The fourth has 400 dollars more than 1/8 ofthe whole,

What is the value of the estate?

Answer 4800 dollars.


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Problem 10. What is that number which is asmuch less than 500, as a fifth part of it isgreater than 40?

Answer 450.


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Problem 11. There are two numbers whosedifference is 40, and which are to each otherratio 6 to 5. What are the numbers?

Answer 240 and 200.


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Problem 12. What number is that, which is to12 increased by three times the number, as 2to 9 ?

Answer 8.


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Problem 13. What number is that, a sixthpart of which exceeds an eighth part of it by20?

Answer 480.


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Problem 14. Divide a prize of 2000 dollarsinto two such parts, that one of them shallbe to the other, as 9:7.

Answer The parts are 1125, and 875.


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Problem 15. What sum of money is that,whose third part, fourth part, and fifth part,added together, amount to 94 dollars?

Answer 120 dollars.


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Problem 16. A man spent one third of his lifein England, one fourth of it in Scotland, andthe remainder of it, which was 20 years, in

the United States. To what age did he live? Answer to the age of 48.


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Problem 17. What number is that 1/4 ofwhich is greater than 1/5 of it by 96?


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Problem 18. What number is that, to which10 being added, 3/5 of the sum will be 66?


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Problem 19. Of the trees in an orchard, 3/4are apple trees, 1/10 pear trees, and theremainder peach trees, which are 20 more

than 1/8 of the whole. What is the wholenumber in the orchard?

Answer 800.


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Problem 20. What number is that, of which if1/3, 1/4, and 2/7 be added together the sumwill be 73?

Answer 84.

unit 1 equation

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