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Algebraic Equations

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 Sub Topics Algebra is the study of mathematics, which helps us to find the value of the unknown variables, which are written in the combinations of the terms. In algebraic equations, we say that the left side and the right side of the equations will be equal in case the correct value of the unknown variable is placed. The value of the variable can be found by moving all the constant values in the equation to one side of the equation so that the variable is left on another side of the equation and thus we are able to get the value of the variable, which satisfies the equation. It contains the terms like numbers, integers, fractions, roots, exponents, ratios, graphing etc. Pre algebra equation is the simple equation which can be solved easily without any complex calculations. Linear equation is an algebraic equation in which each term is either a constant or the product of a constant and with a single variable. It contains one or more variables. It occurs with great regularity in applied mathematics.

Algebraic Equation Definition

Algebraic equation is an equation of the form $A$= $B$ where $A$ and $B$ are polynomials in some field. If a polynomial equation contain several variables then it is called multivariate and the polynomial equation is referred to as algebraic equation. An equation obtained by equating to zero a sum of a finite number of terms is called algebraic equation.
Examples: $x$ $^{2}$ - 5$x$ + 6 = 0, $x$ $^{4}$ - 2$x$ $^{3}$ + 5$x$= 0, $x$ $^{7}$ - 5$x$ $^{4}$ = 0. Linear equations and quadratic equations are the example of the algebraic equations.

Types of Algebraic Equations

Algebraic equations are of different types like

1) Linear Equation: It is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Here the variable expressions have degree one.

2) Quadratic Equation: Quadratic equation is the second degree equation in one variable contains the variable with an exponent of 2.
The standardized form of a quadratic equation is ax$^{2}$ + bx + c = 0 where a, b and c can have any value and 'a' cannot be zero.

3) Cubic Equation: Cubic equation is an equation involving a cubic polynomial.
Standard form of cubic equation is $a_{3}x^{3}$ + $a_{2}x^{2}$ + $a_{1}$x + $a_{0}$ = 0, where 'a' cannot be zero.

4) Polynomial equation of degree n:

A polynomial equation can be written in the form
$f(x)$ = $a_{n}x^{n} + a_{n-1}x^{n-1}+......+ a_{1}x + a_{0} ( a_{n} \neq 0)$
$a_{n}, a_{n-1},......., a_{0}$ are real numbers and n being a non-negative number.

A Polynomial can be expressed in terms that only have positive integer exponents and the operations of addition, subtraction, and multiplication. The above can further be categorized by whether the coefficients in the equations are integer, rational or irrational.

Linear Algebraic Equations

In linear functions the independent variable x will never have an exponent greater than one. A linear function is always represented by a straight line and plotting linear functions is very easy. Any function of the form $f(x)$ = $mx + b$, $m$ $\neq$ 0 is called a linear function. The graph of f is a line with slope '$m$' and $y$ intercept '$b$'. '$m$' is the slope of the line and it controls steepness.

Given below is the graph of a linear function:

Two Step Algebra Equations

Two step algebraic equations can be solved easily. Isolate the variable by using suitable mathematical operations related to the problem.
Given below is an example of two step algebra equation.

Solve: 9$x$ - 7 = - 7

Solution:
The given equation is 9$x$ - 7 = - 7

Step 1 :
Adding 7 to both the sides, we get

=> 9$x$ - 7 + 7 = - 7 + 7

=> 9$x$ = 0

Step 2:
Dividing each side by 9, we get
=> $x$ = 0

Graphing Algebraic Equations

A function's graph makes it very easy to visualize a function and do mathematical calculations on the function. Graphing on a Cartesian plane is known as curve sketching. To determine if a graph of a curve is a function of x, use the vertical line test and to determine if a graph of a curve is a function of y use the horizontal line test. Graph of functions can be linear, quadratic, radical, trignometric, logarithmic etc.,

Consider an example : Graph the solutions of the system: $x + y$ = 5 and $x - y$ = 3

Solution:

Given system: $x + y$ = 5 and $x - y$ = 8

Step 1:
Consider $x + y$ = 5

Plug $x$ = 0 in $x + y$ = 5

=> $y$ = 5

Plug $x$ = 1 in $x + y$ = 5

=> $y$ = 4
Plug $x$ = 4 in $x + y$ = 5

=> $y$ = 1

Solution for $x + y$ = 5

 $X$ 0 1 4 $Y$ 5 4 1

Again, Consider the equation $x - y$ = 3
Plug in $x$ = 0, we get , $y$ = -3
When $x$ = 1, $y$ = -2
When $x$ = 4, $y$ = 1
Solution for $x - y$ = 3
 $X$ 0 1 4 $Y$ -3 -2 1

The obtained points are plotted in the below graph:

Algebraic Equations With Fractions

Given below are the simple steps to be remembered while solving algebraic equations with fractions.
• Choose the common denominator from the fractional equation.
• Multiply each term in the equation by common denominator.
• Form a denominator free equation.
Consider an example: $\frac{x}{5}$ - 2 = 5

Solution :
The given equation is $\frac{x}{5}$ - 2 = 5
Multiplying each term in the given equation by 5, we get
$x$ - 10 = 25
Adding 10 to both the sides, we get
$x$ = 35

Solving Algebraic Equations

Given below are some examples to solve algebraic equations.

Example 1: Solve 5x$^{2}$ + 3x + 1 = 0
Solution :

Step - 1:
Coefficients of $a$, $b$ and c are 5, 3 and 1 respectively.

$x$ = $\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Step - 2 :
Plug in the values of a, b and c in the above formula
$x$ = $\frac{-3\pm\sqrt{3^{2}-4 \times 5 \times 1}}{2 \times 5}$

Step - 3:
Simplify the expression under the square root

$x$ = $\frac{-3\pm\sqrt{-11}}{10}$
Solving for $x$ we get
$x_{1}$ = $\frac{-3+\sqrt{-11}}{10}$

$x_{1}$ = - $\frac{3}{10}$ + $\frac{1}{10}$ $\sqrt{11i}$

and $x_{2}$ = $\frac{-3-\sqrt{-11}}{10}$

$x_{2}$ = - $\frac{3}{10}$ - $\frac{1}{10}$ $\sqrt{11i}$

Step - 4:
Therefore the solutions for the given equation are :

$x_{1}$ = - $\frac{3}{10}$ + $\frac{1}{10}$ $\sqrt{11i}$ and $x_{2}$ = - $\frac{3}{10}$ - $\frac{1}{10}$ $\sqrt{11i}$

Example 2: Solve 5 ($x$ - 4) + 7 (3$x$) + 25 = 15

Solution:

Given: 5 ($x$ - 4) + 7 (3$x$) + 25 = 15

5$x$ - 20 + 21$x$ + 25 = 15

5$x$ + 21$x$ = 10

26x = 10

$x$ = $\frac{10}{26}$

Therefore, $x$ = 0.38

Example 3: Solve the equation x$^{2}$ - 36 = 0

Solution:

$x$ $^{2}$ - 36 = 0

Add 36 on both the sides we get,

$x$ $^{2}$ = 36

Squaring on both sides we get,

$x$ = $\sqrt{6}$

Therefore, $x$ = $\pm$6

Balancing Algebraic Equations

An equation by definition describes a balanced situation where the quantities on one side of the fulcrum are exactly the same as the quantities on the opposite side. If the quantities do not balance then we would not have an equation but an inequality.

Example: 5($x$ + 1) - 4(2$x$) = 20

Solution: 5$x$ + 5 - 8$x$ = 20
5$x$ - 8$x$ = 15
-3$x$ = 15
$x$ = $\frac{-15}{3}$
$x$ = -5
Therefore, $x$ = -5 is the solution to the given equation.

Writing Algebraic Equations

 Sentence Algebraic Equation Twice a number increased is eleven. 2$x$ = 11 Ten less than a number is fifty. $x$ - 10 = 50 A number divided by nine is four. $\frac{x}{9}$ = 4 The quotient of seventy seven and six more than a number is eight. $\frac{77}{x+6}$ = 8 A large bread loaf pie with 15 slices is shared among x students so that each student's share is 7 slices. $\frac{15}{x}$ = 7