वैदिक बीजगणित (11) | बीजगणितीय समीकरण |
(भाग 8.1)
अचर :– संख्या या अंको में व्यक्त राशि को अचर कहते हैं। अचरों का मान अपरिवर्तनशील रहता है। इन्हें हम अंक भी कहते हैं क्योंकि यह अंको के रूप में लिखे जाते हैं – 1, 2, 3, 7, 25, 100, …………
Constant:– The quantity represented by numbers is called constant. These are also called Numerals because these are Numbers. It value is fixed i.e. 1, 2, 3, 7, 25, 100, …………
Numeric Value or counting Numbers are called Constant.
अक्षरों में व्यक्ति राशि को चर कहते हैं । यह अंग्रेजी या रोमन वर्णमाला के हो सकते हैं। इन्हें हम अक्षर भी कहते हैं क्योंकि यह अक्षरों के रूप में लिखे होते हैं। चरों का मान परिवर्तनशील होता है। x, y, z, a, b, c, l, m, n, α, β, γ, θ, λ etc.
Variables: - The quantity represented by letters is called variables. These letters may be from Roman or English alphabets.
These are also called Literals because variables are letters. Its value is not fixed i.e. x, y, z, a, b, c, l, m, n, α, β, γ, θ, λ etc. 2x, 7z, xy, yz.
Alphabets of English and Roman are called Variables.
पद:– एक बीजगणितीय ऐसे नए रूप में लिखा हुआ कुछ पद कहलाता है अर्थात एक चिन्ह वाले मान को पद कहते हैं जैसे ✍x, y, z, a, b, c, l, m, n, 2x, 7z, xy, yz, abc, x², yz², x²y²z²
Term: - Something written algebraically in single SNA form is
called a Term. It is also called Meal. The value of single sign is
known as term i.e. x, y, z, a, b, c, l, m, n, 2x, 7z, xy, yz, abc, x², yz², x²y²z² etc.
चरों की प्रकृति के आधार पर पदों के प्रकार :– चरों की प्रकृति के आधार पर पद दो प्रकार के होते हैं । समान पद और असमान पद।
Types of the terms on the bases of variables: - There are two types of temrs on the bases of variables. Like terms and Unlike tems.
समान पद:– वे पद जिनके समान वर्णिक या अक्षरीय गुणनखंड होते हैं। समान पद कहलाते हैं।
Like Terms : - The terms which have same literal factors is called Like terms i.e. ab and ba are like terms. abc , cab and acb are like tems.
abc , cab and acb are not written according to SNA form if we use it then all three terms will be abc
असमान पद:– वे पद जिनके समान वर्णिक या अक्षरीय गुणनखंड नहीं होते हैं। असमान पद कहलाते हैं।
Unlike Terms: - The terms which have not same literal factors is
called Unlike terms i.e.
02. Find like terms from the following:
Sol. 01 7x, 14x, –13x are like terms.and 5x2 and –9x2 are like
terms
व्यंजक :– पदों के समूह जो गणित के आधारभूत चिन्हों के द्वारा आपस बंधे रहते हैं, व्यंजक कहलाते हैं। अतः एक या उससे अधिक पदों के संयोग से ही कोई व्यंजक बनता है।
पदों के आधार पर व्यंजक कई प्रकार के हो सकते हैं जैसे एक पदीय व्यंजक, द्विपदीय व्यंजक, त्रिपदीय व्यंजक तथा बहुपदीय व्यंजक आदि
Expression: - Group of term connected by the sign of fundamental operations (addition, subtraction, multiplication or division) is known as Expression. The Expression is formed by adding or subtracting one or more terms. These are of many types like monomials, binomials,
trinomials and polynomials.
पदों के आधार पर व्यंजकों के प्रकार:– पदों की संख्या के आधार पर व्यंजक चार प्रकार के होते हैं।
Types of expression on the basses of Number of terms: - There are mainly four types of expressions on the bases of the Numbers of Terms.
एक पदीय व्यंजक:– वह व्यंजक जिसमें केवल एक ही पद (या एक ही चिन्ह) होता है वह एक पदीय व्यंजक कहलाता है। जैसे x, y, z, xyz, ab²c, 2x, 7z, xy, yz, abc, x², yz², x²y²z² etc.
Monomials: - The Expression that contains only one term (or one
sign) is called monomials i.e. x, y, z, xyz, ab²c, 2x, 7z, xy, yz, abc, x², yz², x²y²z² etc.
द्विपदीय व्यंजक:– वह व्यंजक जिसमें केवल एक ही पद (या एक ही चिन्ह) होता है वह एक पदीय व्यंजक कहलाता है। जैसे x + y, z – a, 2x + xyz, a + b²c, 2 + x, z – 2, xy + yz, abc + a²bc, x² + yz², x²+y² etc.
Binomials: - The Expression that contains only two term (or two
signs) is called binomials i.e. x + y, z – a, 2x + xyz, a + b²c, 2 + x, z – 2, xy + yz, abc + a²bc, x² + yz², x²+y² etc. x
त्रिपदीय व्यंजक:– वह व्यंजक जिसमें केवल एक ही पद (या एक ही चिन्ह) होता है वह त्रिपदीय व्यंजक कहलाता है। जैसे ax² + bx + c , x + y + 2, z – a + b, 2x + xy + z, a + b² + c, a + 2 + x, z – 2 – y, xy + yz – zx, abc + a2bc+ bc, x² + y + z², x² + y² + z² etc.
Trinomials: - The Expression that contains only three term (or three signs) is called trinomials i.e.
ax² + bx + c , x + y + 2, z – a + b, 2x + xy + z, a + b² + c, a + 2 + x, z – 2 – y, xy + yz – zx, abc + a2bc+ bc, x² + y + z², x² + y² + z² etc.
बहुपद :- तीन या उससे अधिक पदों (अथवा तीन या अधिक चिन्हों) वाला व्यंजक, जिसके गुणज शून्येत्तर (शून्य न हों) तथा घातें ऋणेत्तर (ऋण न हों) कहलाता है। जैसे
x + y + 2, z – a + b, 2x + xy + z, a + b² + c, a + 2 + x, z – 2 – y, xy + yz – zx, abc + a²bc+ bc, x² + y + z², x²+y²+ z² etc.
Polynomials: - The Expression containing one or more terms ( or
with three or more sign) with non zeros coefficient and with non
negative exponent is known as polynomials i.e.x + y + 2, z – a + b, 2x + xy + z, a + b² + c, a + 2 + x, z – 2 – y, xy + yz – zx, abc + a²bc+ bc, x² + y + z², x²+y²+ z² etc.
Equation
In an equation there is always an equality sign. The equality sign
shows that the value of the expression to the left of the sign (the left hand side or L.H.S.) is equal to the value of the expression to the right of the sign (the right hand side or R.H.S.).
एक समीकरण में, समता या समिका या बराबर का चिन्ह सदैव होता है। समता का चिन्ह यह दर्शाता है कि इस चिन्ह के बाईं ओर के व्यंजक ‘बायाँ पक्ष’ या (LHS) का मान चिन्ह के दाईं ओर के व्यंजक ‘दायाँ पक्ष’ या (RHS) के मान के बराबर है।
Equation:- An equation is a condition on a variable such that two expressions in the variable should have equal value.
OR
In short, an equation is a condition on a variable. The condition is that two expressions should have equal value. Note that at least
one of the two expressions must contain the variable.
| Keyword | Operation |
|---|---|
| is equal | = |
| Add or sum or more or plus | + |
| Subtract or less or minus | – |
| Multiplication or product or times | × |
| divide or quotient | / |
Follow the steps to translate and solve a word problem.
- Read the entire problem several times.
- Look for keywords like more, less, is, etc.
- Replace the keywords with their mathematical operations and shorten the words to variables. Variables can be words or a single letter.
- Solve the resulting equation.
- Answer the problem clearly in a complete sentence.
चार और आचार पदों का योग
Addition of constant and variables
Practice Time 01
01. Write the following statements in the form of equations:
01. The sum of x and 11
02. Add x and 11
03. The sum of numbers x and 11
04. 11 more than x
05. Add 11 with x
06. The sum of p and 2
07. Add p and 2
08. The sum of numbers p and 2
09. 2 more than p
10. Add 2 with p
11. The sum of m and 5
12. Add m and 5
13. The sum of numbers m and 5
14. 5 more than m
15. Add 5 with m
16. The sum of k and 6
17. Add k and 7
18. The sum of numbers k and 2
19. 9 more than k
20. Add 8 with k
चर और अचर पदों के घटा
Subtraction of constant and variables
Practice Time 02
01. Write the following statements in the form of equations:
01. The difference of x and 11
02. Subtract x and 11
03. The difference of numbers x and 11
04. 11 less than x
05. Subtract 11 from x
06. The difference of p and 2
07. Subtract p and 2
08. The difference of numbers p and 2
09. 2 less than p
10. Subtract 2 from p
11. The difference of m and 5
12. Subtract m from 5
13. The difference of numbers m and 5
14. 5 less than m
15. Subtract 5 with m
16. The difference of k and 6
17. Subtract k and 7
18. The difference of numbers k and 2
19. 9 less than k
20. Subtract 8 from k
21. Subtract k from 8
22. Take away 6 from y
23. Take away m from 11
24. Take away 2 from z
25. Take away m from 5
चर और अचर पदों की गुणा
Multiplication of constant and variables
Practice Time 03
01. Write the following statements in the form of equations:
01. The product of x and 11
02. Multiply x by 11
03. The product of numbers x and 11
04. 11 times of x
05. Multiply 11 from x
06. The product of p and 2
07. Multiply p from 2
08. The product of numbers p and 2
09. 2 time of p
10. Multiply 2 from p
11. The product of m and 5
12. Multiply m from 5
13. The product of numbers m and 5
14. 5 Times of m
15. Multiply 5 by m
16. The product of k and 6
17. Multiply k by 7
18. The product of numbers k and 2
19. 9 times of k
20. Multiply 8 from k
21. Multiply k from 8
22. Double of m
23. Double of p
24. Triple of m
25. One fourth of m
26. One third of k
27. Two third of n
28. Three fourth of k
29. One third of z
30. Four times of k
चर और अचर पदों की भाग
Division of constant and variables
Practice Time 04
01. Write the following statements in the form of equations:
01. Divide x from 11
02. Divide x by 11
03. Divide 11 from z
04. Divide z by 2
05. Divide 5 from x
06. Divide p from 2
07. Divide 2 from p
08. The number b divided by 5
09. p divided by 7
10. The number m divided by 2
11. Divide m from 6
12. Divide z by 2
13. Divide 9 from z
14. Divide z by 5
15. Divide 5 from z
16. Divide p from 4
17. Divide 3 from p
18. The number m divided by 5
19. p divided by 8
20. The number m divided by 4
Practice Time 05
Convert the following equations in statement form:
(01) p + 4 = 15
(02) x + 4 = 13
(03) x + 5 = 18
(04) x + 10 = 12
(05) x + 1 = 0
(06) x + 6 = 2
(07) y + 4 = 4
(08) y + 4 = – 4
(09) M + 4 = 43
(10) z + 4 = – 5
(11) 4 + p = 15
(12) 4 + x = 13
(13) 5 + m = 18
(14) 10 + n = 12
(15) 1 + m = 0
(16) 6 + x = 2
(17) 3 + y = 4
(18) 4 + p = – 4
(19) 4 + M = 4
(20) 2 + z = – 5
Practice Time 06
Convert the following equations in statement form:
(01) m – 7 = 3
(02) y – 5 = 7
(03) x – 10 = 14
(04) x – 1 = 0
(05) x – 1 = 5
(06) y – 4 = – 7
(07) y – 4 = 4
(08) m – 5 = 3
(09) y – 8 = 7
(10) x – 1 = 14
(11) x – 5 = 0
(12) x – 6 = 5
(13) M – 2 = – 7
(14) P – 4 = 2
(15) x – 7 = 5
(16) y – 4 = – 7
(17) y – 4 = 4
(18) m – 5 = 3
(19) y – 8 = 7
(20) x – 1 = 14
Practice Time 07
Convert the following equations in statement form:
A
(01) 2 × m = 14
(02) 3 × n = 12
(03) 6 × x = 18
(04) 7 × y = 14
(05) 2 × x = 6
(06) 4 × m = 16
(07) 7 × y = 10
(08) 4 × y = 16
(09) 5 × m = 25
(10) 7 × n = 35
(11) 4 × x = 25
(12) 8 × y = 36
(13) 6 × x = 25
(14) 4 × y = 36
(15) 3 × x = 24
(16) 8 × y = 48
B
(01) 2 × m + 4 = 8
(02) 4 × m – 5 = 11
(03) 11 + 2 × a = 25
(04) 12 × x + 4 = 3
(05) 75 – 3 × a =35
(06) 2 × x – 3 = 5
(07) 7 – 4 × y = 3
(08) 7 × y – 10 = 0
(09) 4 × y – 16 = 0
(10) 5 × m – 25 = 0
(11) 5 × m + 45 = 0
(12) 3 × x + 9 = 0
(13) 2 × x – 6 = 0
(14) 3 × p + 4 = 25
(15) 4 × p – 2 = 18
08. 20 x = x + 95
(04) 12 × x + 4 = 3 × x
Practice Time 08
Convert the following equations in statement form:
(01) m/6 = 3
(02) m/2 = 6
(03) x/ 3 = 6
(04) y/6 = 7
(05) y/8 = 4
(06) b/2 = 8
(07) p/7 = 4
(08) z/3 = 5 /4
(09) a /5 = 7/15
(10) p/2 + 2 = 8
(11) p/2 – 2 = 8
(12) x/2 + 5 = 10
(13) x/8 + 6 = 8
(14) 3m/6 = 6
(15) 2x/ 3 = 6
(16) 3b/2 = 3
(17) 2p/7 = 4
(18) 5z/3 = 5 /4
(19) 7a /5 = 7/15
(20) 3p/2 + 2 = 8
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