ALGEBRA
Category : 7th Class
Learning Objectives:
SIMPLE EQUATION
EQUATION:
A statement of inequality which contain one or more unknown quantities or variables is known as equation.
Example: \[x+4=11\]
ROOT OF THE EQUATION OR SOLUTION:
A number which satisfies an equation is called the solution or root of the equation.
Example: In \[5+x=7,\]\[2\] is the solution or root of the equation.
If we want to check it,
\[5+x=7\] \[x=2\]
\[5+2=7\]
\[7=7\]
SOLVING SIMPLE EQUATION:
To solve the simple equations the following facts should be remember.
Example: Solve an equation \[x+4=12\]
Solution: \[x+4=12\]
\[x+4-4=12-4\]
\[x=8\]
Explanation: We want only\[x\]to remain on the left side to have this, we will have to subtract 4. To keep the equation balanced, we will have to subtract 4 from right side also.
TRANSPOSITION:
Any term of an equation may be taken from one side to the other with a change in its sign. This does not affect the equality of equation. This process is called transposition.
Example: Solve\[5x-6=4x-2\]
Solution: \[5x-6=4x-2\]
\[\Rightarrow \] \[5x-4x=-2+6\] [Transposing \[4x\] to LHS and\[-6\]to RHS]
\[\Rightarrow \] \[x=4\]
Check: Substituting\[x=4\]in the given equation, we get
\[\text{LHS}=5\times 4-6=20-6=14,\]and
\[\text{RHS}=4\times 4-2=16-2=14\]
\[\therefore \] \[\text{LHS = RHS}\]
Equation in which variable appear on both sides:
Example: \[5x+4=3x+20\]
Solution: \[5x+4=3x+20\] we want to subtract\[3x\]from right side, we can do so by subtracting \[3x\] from both sides.
\[5x+4-3x=3x+20-3x\]
\[2x+4=20\] we want to remove 4 from LHS we can do so by subtracting 4 from both sides.
\[2x+4-4=20-4\]
\[2x=16\]Dividing both sides by 2.
\[\frac{2x}{2}=\frac{16}{2}\]
\[x=8\]
Forming Simple Equation:
Example: When you add 3 to one-third of z, you get 30. Find the value of z.
Solution: \[\frac{z}{3}+3=30\]
\[\frac{z}{3}=30-3\]
\[\frac{z}{3}=27\]
\[z=27\times 3\]
\[=81\]
Example: Three times a number is equal to 24. Find the number.
Solution: Let the number\[=x\]
Then, \[3x=24\]
\[x=\frac{24}{3}\Rightarrow 8\]
Hence, required number is 8.
ALGEBRAIC EXPRESSION
A combination of constants and variables connected by some or all of the four fundamental operations \[+,\,\,-,\,\,\times \] and\[\div \]is called an algebraic expressions.
CONSTANT:
A symbol having fixed numerical value is called constant.
VARIABLE:
A symbol which can have various numerical value is called variable.
Example:
(1) The perimeter P of a square of side S is given by the formula P = 4\[\times \]S. Where P and S are variables and 4 is constant.
(2) The circumference C of a circle have radius R is given by the formula\[\text{C}\,\text{=}\,\text{2 }\!\!\pi\!\!\text{ R}\text{.}\]Where C and R are variable while 2 and\[\text{ }\!\!\pi\!\!\text{ }\]are constant.
TERMS OF AN ALGEBRAIC EXPRESSION:
The different parts of algebraic expression separated by the sign \[+\] or \[-,\] are called the terms of algebraic expression.
Example: \[3+2xy+5{{x}^{2}}y\] have three terms \[3,\]\[2xy\] and \[5{{x}^{2}}y\]
VARIOUS TYPES OF ALGEBRAIC EXPRESSIONS:
(i) Monominals: An algebraic expression which contain only one term, is called monominals.
Examples: \[5x,\,\,2y,\,\,3xy,\,\,-3{{a}^{2}},\,\,-5{{a}^{2}}b\] etc. are monominals.
(ii) Binominals: An algebraic expression which contain two terms is called binominals.
Examples: \[5x+y,\,\,2y+z,\,-3{{x}^{2}}+{{y}^{2}},\,\,2{{a}^{2}}+ab\] etc. are binominals.
(iii) Trinominals: An algebraic expression contain three terms are called trinominals.
Examples: \[\left( a+2b+5c \right),\left( {{x}^{2}}+xy+6{{y}^{2}} \right),\left( {{x}^{2}}-{{y}^{3}}-{{z}^{3}} \right)\]etc. are trinominals.
(iv) Quadrinominals: An algebraic expression containing four terms are called quadrinominals.
Example:\[\left( 2x+y+3z+5 \right),\]\[\,\left( {{x}^{2}}+2{{y}^{2}}+zz+xz \right),\,\]\[\left( {{x}^{3}}+{{y}^{3}}+{{z}^{3}}+3xy \right)\]etc. are quadrinominals.
(v) Polynominals: An expression containing two or more term is called a polynominal.
FACTORS:
Each term of algebraic expression consists of a product of constants and variables. A constant factor is called numerical factor, while variable factor is known as a literal factor.
COEFFICIENT:
In a term of an algebraic expression any of the factors with the sign of the term is called the coefficient of the product of the other factors in that term.
Example:
(1) In \[3ab,\] the coefficient of \[a\] is \[3b\] and the coefficient of \[b\] is \[3a.\]
(2) In \[-\,4xy,\] the coefficient of \[x\] is \[-\,4y\] and the coefficient of \[y\] is \[-\,4x.\]
Constant term: A term of expression having no literal factor is called a constant term.
Example: In the algebraic expression \[2x+3y-3\] constant term is \[-3.\]
Like and unlike terms: In a given algebraic expression, the terms having the same literal factors are called like or similar terms, otherwise they are called unlike terms.
Addition of algebraic expressions: Write each expression in a separate row in such a way that like terms are arranged one below the other in a column. Then add each column.
Example:
(i) Add \[2x+3y\] and \[3x+4y\]
Solution:
\[\frac{\begin{align}
& 2x+4y \\
& 3x+4y \\
\end{align}}{5x+7y}\]
(ii) Add \[8{{x}^{2}}-7{{y}^{2}}\] and \[-8{{x}^{2}}+7{{y}^{2}}\]
Solution:
\[\frac{\begin{align}
& 8{{x}^{2}}-7{{y}^{2}} \\
& -8{{x}^{2}}+7{{y}^{2}} \\
\end{align}}{0\,+\,0\,=0}\]
SUBTRACTION OF ALGEBRAIC EXPRESSIONS:
(1) To subtract simple expressions:
(a) To subtract similar terms, change the sign of the subtrahend and proceed as addition.
(b) The subtraction of terms which are not similar can only be indicated. However, the sign of the subtrahend must be changed.
Example:
(i) Subtract \[-7xyz\] from \[2xyz\]
Solution:
\[\begin{align}
& \underline{\begin{align}
& \,\,\,\,\,2xyz \\
& -\,7xyz \\
& (+)\,(Change\,sign) \\
\end{align}} \\
& \,\,\,\,\,9xyz \\
\end{align}\]
(ii) Subtract \[8\text{ }abc\] from \[10\text{ }abc\]
Solution:
\begin{align}
& \underline{\begin{align}
& +\,\,10\text{ }abc \\
& +\,\,08\,\,abc \\
& (-)\,change\,sign \\
\end{align}} \\
& \,\,\,\,\,2abc \\
\end{align}
(2) To subtract compound expressions: If the polynominal are so arranged that similar terms are in vertical columns change the sign of each term of the subtrahend and proceed as in addition.
Example:
(i) Subtract \[\left( 2x+3y \right)\] from \[\left( 3x+7y \right)\]
Solution:
\begin{align}
& \underline{\begin{align}
& \,\,3x+7y \\
& \,\,2x+3y \\
& \,\,(-)\,\,(-) \\
\end{align}} \\
& \,\,x+4y \\
\end{align}
(ii) Subtract \[-\,2a+b+4d\] from \[4a-2b-c\]
Solution:
\begin{align}
& \underline{\begin{align}
& \,\,\,\,4a-2b-c \\
& -\,2a+b+4d \\
& (+)\,\,\,\,\,\,(-)\,\,\,\,\,\,\,\,\,\,(-) \\
\end{align}} \\
& 6a-3b-c-4d \\
\end{align}
MULTIPLICATION OF ALGEBRAIC EXPRESSIONS:
Rule 1. The product of two factors with like signs is positive, and the product of two factors with unlike signs is negative.
Rule 2. If a is any variable and w, n are positive integers then \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
Example: \[{{x}^{3}}\times {{x}^{2}}={{x}^{3+2}}={{x}^{5}}\]
Rule 1: The coefficient of the product of two monomials is equal to the product of their coefficients.
Rule 2: The variable part in the product of two monomials is equal to the product of the variables, in the given monomials.
These rules may be extended for the product of three or more monomials.
Example:
(i) \[5ab\times 2b\]
\[=(5\times 2)\times (a\times b\times b)\]
\[=10\,a{{b}^{1+1}}\]
\[=10\,a{{b}^{2}}\]
(ii) \[3a{{b}^{2}}{{c}^{3}}\times 5{{a}^{2}}bc\]
\[(3\times 5)\times ({{a}^{2}}\times {{b}^{2}}\times a\times b\times {{c}^{3}}\times c)\]
\[15\times ({{a}^{1+2}}\times {{b}^{2+1}}\times {{c}^{3+1}})\]
\[15\,\,{{a}^{3}}{{b}^{3}}{{c}^{4}}\]
Let \[x,\,\,y\] and \[z\] be three monomials.
Then, by distributive law of multiplication over addition, we have:
\[a\times \left( b+c \right)=\left( a\times b \right)+\left( a\times c \right)\]
Examples: Multiply \[\left( 4xy+3y \right)\] by \[4xy\]
Solution: \[4xy\times \left( 4xy+3y \right)\]
\[=\left( 4xy\times 4xy \right)+\left( 4xy+3y \right)\]
\[=16{{x}^{2}}{{y}^{2}}+12x{{y}^{2}}\]
Suppose\[(a+b)\]and\[(a+b)\]are two binominals. By using the distributive law of multiplication over addition twice, we may find their product as given below:
\[\left( a+b \right)\times \left( c+d \right)=a\times \left( c+d \right)+b\times \left( c+d \right)\]
\[=\left( a\times c+a\times d \right)+\left( b\times c+b\times d \right)\]
\[=ac+ad+bc+bd\]
Example: Multiply\[(3{{x}^{2}}+{{y}^{2}})\] by \[(2{{x}^{2}}+3{{y}^{2}})\]
Solution: \[(3{{x}^{2}}+{{y}^{2}})\times (2{{x}^{2}}+3{{y}^{2}})\]
\[(3{{x}^{2}}\times 2{{x}^{2}})+(3{{x}^{2}}\times 3{{y}^{2}})+({{y}^{2}}\times 2{{x}^{2}})+({{y}^{2}}\times 3{{y}^{2}})\]\[6{{x}^{4}}+9{{x}^{2}}{{y}^{2}}+2{{x}^{2}}{{y}^{2}}+3{{y}^{4}}\]
\[6{{x}^{4}}+11{{x}^{2}}{{y}^{2}}+3{{y}^{4}}\]
Standard Identities:
(i) \[{{(a+b)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]
(ii) \[{{(a-b)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\]
(iii) \[({{a}^{2}}-{{b}^{2}})=(a+b)\,(a-b)\]
(iv) \[(x+a)\,(x+b)={{x}^{2}}+(a+b)x+ab\]
(v) \[{{(a+b+c)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ca\]
Example: Factorize\[={{x}^{2}}+6x+9\]
Solution:
By observation we find the x2 and 9 are perfect square terms preceded by plus sign and 6x can be expressed as \[2\,(3)\,(x)\] which is twice the product of square root of perfect square terms
thus \[{{x}^{2}}+6x+9={{x}^{2}}+2x.3+{{3}^{2}}\]
\[{{x}^{2}}+6x+9={{(x+3)}^{2}}=(x+3)\,(x+3)\] [using identity \[{{(a+b)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]]
Example: Factorize: \[16{{b}^{2}}-40by+25{{y}^{2}}\]
Solution:
\[16{{b}^{2}}-40by+25{{y}^{2}}={{(4b)}^{2}}-2(4b)\,(5y)+{{(5y)}^{2}}={{(4b-5y)}^{2}}\]
\[16{{b}^{2}}-40by+25{{y}^{2}}=(4b-5y)\,(4b-5y)\][using identity \[{{a}^{2}}-2ab+{{b}^{2}}={{(a-b)}^{2}}\]]
Factorizing Difference of Two Squares:
Algebraic expression in the form of difference of two squares is factorized as follows:
(a) Find square roots of two square terms.
(b) Write sum of the two square roots as one of the factors and difference of two square roots as the other factor.
Example: Factorize: \[\frac{36}{49}{{p}^{2}}-81\]
Solution: \[\frac{36}{49}{{p}^{2}}-81={{\left( \frac{6}{7}p \right)}^{2}}-{{(9)}^{2}}\]
\[\frac{36}{49}{{p}^{2}}-81=\left( \frac{6}{7}p+9 \right)\left( \frac{6}{7}p-9 \right)\]
EXPONENTS AND POWERS
When a number is multiplied itself by several times it can be written in short form as:
\[3\times 3\times 3\times 3\times 3={{3}^{5}}\]
Number is known as base and number of times is known as exponent.
For any rational number a and positive integer\[n,\]
\[{{a}^{n}}=a\times a\times a\times ....\times a(n\,\,\text{times})\]
\[{{a}^{n}}\] is known as \[{{\text{n}}^{\text{th}}}\] power of a or a raised to the power\[n\]
Example:
(i) \[4\times 4\times 4\times 4\times 4={{4}^{5}}\]
(ii) \[\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}={{\left( \frac{1}{2} \right)}^{4}}\]
Note: Power 2 and 3 have special names as:
Power 2 is known as SQUARE
Power 3 is known as CUBE
Some more examples:
|
Expression |
Meaning |
Exponent |
Base |
Value |
|
\[{{3}^{4}}\] |
\[3\times 3\times 3\times 3\] |
\[4\] |
\[3\] |
\[81\] |
|
\[{{(-4)}^{3}}\] |
\[-4\times -4\times -4\] |
\[3\] |
\[-4\] |
\[-64\] |
|
\[{{\left( \frac{1}{2} \right)}^{3}}\] |
\[\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\] |
\[3\] |
\[\frac{1}{2}\] |
\[\frac{1}{8}\] |
|
\[{{\left( -\frac{2}{3} \right)}^{4}}\] |
\[-\frac{2}{3}\times -\frac{2}{3}\times \]\[-\frac{2}{3}\times -\frac{2}{3}\] |
\[4\] |
\[-\frac{2}{3}\] |
\[\frac{16}{81}\] |
Remember:
LAWS OF EXPONENTS:
Law 1. If \[\frac{a}{b}\] is any rational number and m and n are positive integers, then \[{{\left( \frac{a}{b} \right)}^{m}}\times {{\left( \frac{a}{b} \right)}^{n}}={{\left( \frac{a}{b} \right)}^{m+n}}\]
Example:
(i) \[{{\left( \frac{-3}{4} \right)}^{2}}\times {{\left( \frac{-3}{4} \right)}^{3}}={{\left( \frac{-3}{4} \right)}^{2+3}}={{\left( \frac{-3}{4} \right)}^{5}}\]
(ii) \[{{\left( \frac{5}{8} \right)}^{5}}\times {{\left( \frac{5}{8} \right)}^{3}}={{\left( \frac{5}{8} \right)}^{5+3}}={{\left( \frac{5}{8} \right)}^{8}}\]
Law 2. If \[\frac{a}{b}\] is any rational number and m and n are two positive integers as \[m>n,\] then\[{{\left( \frac{a}{b} \right)}^{m}}\div {{\left( \frac{a}{b} \right)}^{n}}={{\left( \frac{a}{b} \right)}^{m-n}}\]
Example:
(i) \[{{\left( -\frac{3}{4} \right)}^{4}}\div {{\left( -\frac{3}{4} \right)}^{2}}={{\left( \frac{-3}{4} \right)}^{4-2}}={{\left( \frac{-3}{4} \right)}^{2}}\]
(ii) \[{{(4)}^{8}}\div {{(4)}^{6}}={{(4)}^{8-6}}={{(4)}^{2}}\]
Law 3. If \[\frac{a}{b}\] is any rational number and \[m\] and \[n\] are two positive integers as \[m<n,\]then
\[{{\left( \frac{a}{b} \right)}^{m}}\div {{\left( \frac{a}{b} \right)}^{n}}=\frac{{{\left( \frac{a}{b} \right)}^{m}}}{{{\left( \frac{a}{b} \right)}^{n}}}=\frac{1}{{{\left( \frac{a}{b} \right)}^{n-m}}}\]
Example:
(i) \[\frac{{{(2)}^{7}}}{{{(2)}^{11}}}=\frac{1}{{{(2)}^{11-7}}}=\frac{1}{{{(2)}^{4}}}\]
(ii) \[{{\left( \frac{4}{5} \right)}^{3}}\div {{\left( \frac{4}{5} \right)}^{5}}=\frac{1}{{{\left( \frac{4}{5} \right)}^{5-3}}}=\frac{1}{{{\left( \frac{4}{5} \right)}^{2}}}={{\left( \frac{5}{4} \right)}^{2}}\]
Law 4. If \[\frac{a}{b}\] is any rational number and m and n are two positive integers, then \[{{\left\{ {{\left( \frac{a}{b} \right)}^{m}} \right\}}^{n}}={{\left( \frac{a}{b} \right)}^{mn}}\]
Examples:
(i) \[{{\left\{ {{\left( \frac{2}{5} \right)}^{3}} \right\}}^{2}}={{\left( \frac{2}{5} \right)}^{3\times 2}}={{\left( \frac{2}{5} \right)}^{6}}=\frac{{{2}^{6}}}{{{5}^{6}}}=\frac{64}{15625}\]
(ii) \[{{\left\{ {{\left( \frac{-3}{2} \right)}^{2}} \right\}}^{-3}}={{\left( -\frac{3}{2} \right)}^{2\times -3}}={{\left( \frac{-3}{2} \right)}^{-6}}={{\left( \frac{2}{-3} \right)}^{6}}\]
\[=\frac{{{2}^{6}}}{{{(-3)}^{6}}}=\frac{64}{729}\]
Law 5. If \[\frac{a}{b}\] is any rational number and m and n are two positive integers, then
\[{{\left( \frac{a}{b} \right)}^{-n}}=\frac{1}{{{\left( \frac{a}{b} \right)}^{n}}}=\frac{1}{\frac{{{a}^{n}}}{{{b}^{n}}}}=\frac{{{b}^{n}}}{{{a}^{n}}}={{\left( \frac{b}{a} \right)}^{n}}\]
Thus, \[{{\left( \frac{a}{b} \right)}^{-n}}={{\left( \frac{b}{a} \right)}^{n}}\]
Examples:
(i) \[{{\left( \frac{6}{5} \right)}^{-3}}={{\left( \frac{5}{6} \right)}^{3}}=\frac{{{(5)}^{3}}}{{{(6)}^{3}}}=\frac{125}{216}\]
(ii) \[{{\left( \frac{3}{8} \right)}^{-2}}={{\left( \frac{8}{3} \right)}^{2}}=\frac{64}{9}\]
Law 6. If \[\frac{a}{b}\] is any rational number and m and n are two positive integers, then \[{{\left( \frac{a}{b} \right)}^{\circ }}=1\]
Examples:
(i) \[{{\left( \frac{3}{8} \right)}^{\circ }}=1\]
(ii) \[{{\left( \frac{23}{25} \right)}^{\circ }}=1\]
Law 7. If \[\frac{a}{b}\] is any rational number and m and n are two positive integers, then \[{{\left( \frac{a}{b} \right)}^{-1}}=\left( \frac{b}{a} \right)\]
Examples:
(i) \[{{\left( \frac{9}{4} \right)}^{-1}}\]
\[{{\left( \frac{9}{4} \right)}^{-1}}=\frac{4}{9}\]
(ii) \[{{\left( \frac{28}{19} \right)}^{-1}}\]
\[{{\left( \frac{28}{19} \right)}^{-1}}=\frac{19}{28}\]
(iii) \[{{\left( \frac{13}{7} \right)}^{-1}}\]
\[{{\left( \frac{13}{7} \right)}^{-1}}=\frac{7}{13}\]
EXPRESSING LARGE NUMBERS IN STANDARD FORM:
A number is said to be in standard form if it can be expressed as \[k\times {{10}^{n}},\] where \[k\] is a real number such that \[1\le k<10\] and\[n\]is positive integer.
Examples: Express the following in standard form:
(i) \[3859000\]
(ii) \[987500000\]
(iii) \[683000000\]
Solution:
(i) \[3859000=3.859\times {{10}^{6}}\]
(ii) \[987500000=9.875\times {{10}^{8}}\]
(iii) \[683000000=6.83\times {{10}^{8}}\]
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