How to Solve One Variable Equations

Solution of Equation in One Variable

Introduction:
The equation of the form f(x)=0 is called an equation in one variable. It may be algebraic or transcendental or a combination of both. When f(x) is a polynomial in x, that is f(x)=a₀xⁿ+a₁x^n-1++a₂x^n-2+ ..... +a_n then f(x)=0 is called algebraic equation, Again if f(x) contains some other functions such as logarithmic, exponential, trigonometric etc, then f(x)=0 is called transcendental equation.

Example:

A few one variable equation:

(i) x³-2x-5=0

(ii) x³+x²-1=0

(iii) x²-3=0 etc.

A few one variable equation:

(i) e^-x=sinx

(ii) e^x tanx=1

(iii) 2x=log₁₀x+7 etc.

Definitions and Concepts of Sets, Relations and Functions

If  f(x)=0 is an equation and f(a)=0 then x=a is called a root of the equation. Again if the graph of y=f(x) crosses the x-axis at x=a then x=a is a root of the equation f(x)=0. Sometimes root of equation is called zero or solution of the equation.

Mathematical Example:

f(x)=x³-6x²+11x-6=0 is an equation and f(1)=1-6+11-6=0, So x=1 is a root of the equation. Similarly x=2 and x=3 are roots of the equation.

The graph of f(x)=x³-6x²+11x-6 crosses the x-axis at (-2, 0) and (3,0). So x=-2 and x=3 are the roots of the equation f(x)=0.

Graphical Method

Discuss the Graphical method to find a real root of an equation f(x)=0 

Solution: If we take a set of rectangular coordinate axes and plot the graph of y=f(x), then the abscissas of the point where the graph crosses the x-axis are the real roots of the equations f(x)=0.

In most cases the approximate values of the real roots of f(x)=0 are most easily found by writing the equation in the form f₁(x)=f₂(x) and then plotting the two equations y=f₁(x) and y=f₂(x) on the same axes. The abscissas of the point of intersection of these two functions y=f₁(x) and y=f₂(x) are the real roots of the equation f(x)=0.

Problem: Find the approximate value of a root of the equation cosx=3x-1 usnig graphical method.

Solution:

Given equation, cosx=3x-1

Let, y=cosx ............................ (i)

and y=3x-1 ............................ (ii)

Now we plot (i) and (ii) separately on the same set of axes as shown on the following diagram.

The abscissa of the point of intersection not the graphs of theses equations is seen to be about 0.6. hence the approximate value of the root is 0.6.