Linear algebraic equations with two variables

Linear equations with two variables are as follows:

15x + 7y = 52

To solve such a system of simultaneous equations such as

15x + 7y = 52-----------------------1
5x + 3y = 18------------------------2

We can do as follows:

Method-1:
After Multiplying equation-2 with "3", we will get


15x + 7y = 52-----------------------1
3x(5x + 3y = 18)------------------------3

=>

15x + 7y = 52-----------------------1
15x + 9y = 54------------------------3

By subtracting these two equation, we get
 15x + 7y =  52
-15x - 9y = -54
-----------------
0 -2y = -2
=> y = -2/-2
=> y = 1

By putting this value of "y" in equation-1, we get
15x + 7y = 52
=> 15x + 7(1) = 52
=> 15x = 52-7
=> 15x = 45
=> x = 45/15 = 3

So, x = 3

Method-2:

15x + 7y = 52-----------------------1
5x + 3y = 18------------------------2

Take equation-2
5x + 3y = 18
=> x = (18-3y)/5

By putting this value of "x" in equation-1, we will get

15((18-3y)/5) + 7y = 52
=> 54-9y + 7y = 52
=> 54-2y = 52
=> -2y =-2
=> y = 1

By putting this value of "y" in equation-2, we get

x = (18-3)/5 = 15/5 = 3
=> x = 3

Further Reading:
Cracking the GRE, 2010 Edition (Graduate School Test Preparation)

Kaplan GRE Exam Vocabulary in a Box

Barron's GRE

Kaplan GRE Exam 2010-2011 Premier with CD-ROM (Kaplan GRE Premier Program (W/CD))

Kaplan GRE Exam 2010: Strategies, Practice, and Review

Crash Course for the GRE, 3rd Edition (Graduate School Test Preparation)

Linear algebraic equations with one variable

Equations having one variable are easy to solve in finding the value of that variable.

Equivalent equations having the same solution, for example

x + 5 = 15,
3x+15=45
are two equivalent equations.

Both of these equations are true when the value of x is 10 and otherwise these are false equations.

Further Reading:
Cracking the GRE, 2010 Edition (Graduate School Test Preparation)

Kaplan GRE Exam Vocabulary in a Box

Kaplan GRE Exam 2010-2011 Premier with CD-ROM (Kaplan GRE Premier Program (W/CD))

GRE: Practicing to Take the General Test 10th Edition (Practicing to Take the Gre General Test)

Barron's GRE with CD-ROM

Absolute value

The absolute value of any number "x" is denoted by |x| such that

|2| = 2 , |0| = 0

and |-4| =-(-4) = 4

It is to be noted that absolute value of a number cannot be negative.

Further Reading:
Cracking the GRE, 2010 Edition (Graduate School Test Preparation)

Kaplan GRE Exam Vocabulary in a Box

Barron's GRE

Essential Words for the GRE (Barron's Essential Words for the GRE)

Kaplan GRE Exam 2010-2011 Premier with CD-ROM (Kaplan GRE Premier Program (W/CD))

Call (Towards right path)

You are the best nation ever to be produced before mankind. You enjoin the right, forbid the wrong and believe in Allah. If the People of the Book were to believe, it would be better for them. Some of them are believers but most of them are deviators. (110)
 (Holy Quran, Surah Al ‘Imran (The Family Of 'Imran, The House Of 'Imran), Surah # 3: Ayah # 110)

Let there be a community among you who call to the good, and enjoin the right, and forbid the wrong. They are the ones who have success. (104)
(Holy Quran, Surah Al ‘Imran (The Family Of 'Imran, The House Of 'Imran), Surah # 3: Ayah # 104)

Make allowances for people, command what is right, and turn away from the ignorant. (199) If a suggestion from Satan assail thy (mind), seek refuge with Allah. for He heareth and knoweth (all things). (200)
(Holy Quran, Surah Al-A’araf (The Heights), Surah # 7: Ayah # 199-200)

Restrain yourself patiently with those who call on their Lord morning and evening, desiring His face. Do not turn your eyes from them, desiring the attractions of this world. And do not obey someone whose heart We have made neglectful of Our remembrance and who follows his own whims and desires and whose life has transgressed all bounds. (28)
(Holy Quran, Surah Al-Kahf (The Cave), Surah # 18: Ayah # 28)

(Believers are) those who, when they are reminded of the Signs of their Lord, do not turn their backs, deaf and blind to them; (73)
(Holy Quran, Surah Al-Furqan (The Criterion), Surah # 25: Ayah # 73)

Do not let them debar you from Allah’s Signs after they have been sent down to you. Call people to your Lord and on no account be one of the idolaters. (87)
(Holy Quran, Surah Al-Qasas (The Narrations), Surah # 28: Ayah # 87)

(Luqman said,) ‘My son, perform prayer and command what is right and forbid what is wrong and be steadfast in the face of all that happens to you. That is certainly the most resolute course to follow. (17)
(Holy Quran, Surah Luqman (Luqman), Surah # 31: Ayah # 17)

O ye who believe fear ALLAH, and (always) say a word directed to the Right: (70)
(Holy Quran, Surah Al-Ahzab (The confederates), Surah # 33: Ayah # 70)

So call upon Allah, making your religion sincerely His, even though the disbelievers detest it.
(14)
(Holy Quran, Surah Ghafir (Forgiver) or Al-Mu’min (The believer), Surah # 40: Ayah # 14)

If an evil urge from Satan eggs you on, seek refuge in Allah. He is the All-Hearing, the All- Knowing. (36)
(Holy Quran, Surah Fussilat (Expounded) or HA MIM, Surah # 41: Ayah # 36)

Brother

If two parties of the believers fight, make peace between them. But if one of them attacks the other unjustly, fight the attackers until they revert to Allah’s command. If they revert, make peace between them with justice, and be even-handed. Allah loves those who are evenhanded. (9) The believers are brothers, so make peace between your brothers and have fear of Allah so that hopefully you will gain mercy. (10)

(Holy Quran, Surah Al-Hujurat (The Chambers), Surah # 49: Ayah # 9-10)

Betray

You who believe! Do not betray Allah and His Messenger, and do not knowingly betray your trusts. (27)

(Holy Quran, Surah Al-Anfal (The Spoils of War), Surah # 8: Ayah # 27)

Balance

Give full measure when you measure and weigh with a level balance. That is better and gives the best result. (35)

(Holy Quran, Surah Al Isra’ (The Night Journey) or Bani Isra’il (The Children of Israel), Surah # 17: Ayah # 35)

In order that ye may not transgress (due) balance. (8) Give just weight–do not skimp in the balance. (9)

(Holy Quran, Surah Al-Rahman (The Most Gracious), Surah # 55: Ayah # 8-9)