- Dec 16, 2020
- Blog English, Competitive Exam,
Solving algebraic equations is a skill set that is crucial for all competitive examinations. One must master the technique of solving algebraic equations to ace any competitive exam. Algebraic Equations are a part of General Mathematics which is common in all competitive exams like CAT, CLAT, RAS, IAS, Bank exams, and many more.
To know how to solve algebraic equations easily, one must first be aware of what is an Algebraic Equation.
An Algebraic Equations denotes that the value on the left of the equals to sign is equivalent to the value on the right. In simple words an algebraic equation says that two values are equal. Some examples of algebraic equations are given below:
x + 4 = 10
2x + 7y -8 = 4y
3x + 5 = 6
x+ y = 10
x + 2y + 3x = 29
(x−9)(x−7) = 63
An algebraic equation consists of two entities, a variable and a constant. The variable is denoted by alphabets or symbols while a constant is a number. Every algebraic equation has a solution. A solution is a value that you put in the variable to satisfy the equation. For example, consider an equation,
X + 10 = 14
Here, if X = 4, the equation will be satisfied or will be true. Therefore, the solution of this equation is 4.
- X= 4
An equation can also have more than one solution.
For example, consider the equation,
(x−3)(x−2) = 0
When the value of x is 3, we get:
(3−3)(3−2) = 0 × 1 = 0
which satisfies the equation;
And when the value of x is 2, we get:
(2−3)(2−2) = (−1) × 0 = 0
which also satisfies the equation;
So, the solutions are:
x = 3, or x = 2
We combine all the solutions to make a Solution Set
How to Solve an Algebraic Equation?
Solving an Equation with one variable:
- The first step to solve any algebraic equation is to write the problem down.
- Let’s say we are working on a problem 4x + 7 = 27.
- The next step is to isolate the variable using either of the four – Addition, Subtraction, Multiplication, or Division.
- Here for this problem, we will move the constant +7 to the RHS (Right Hand Side).
- Or you can subtract the constant on both the sides.
- This narrows the equation to 4x +7 – 7 = 27 – 7
=> 4x = 20
- Now, to remove the multiple of the variable, we the divide both the sides with 4
- 4x/4 = 20/4
=> x = 5
Solving an Equation with one variable on each side:
- Consider the equation 4x -17 = 19 – 2x
- Since the variable on both the sides are same, start with moving the variables on one side and the constants on the other.
- This narrows the equation to 4x + 2x = 19 + 17
- Since both the expressions on the LHS have the same variable, they can be added.
- This gives us 6x = 19 + 17
=> 6x = 36
- Now, dividing both the sides by the multiple of the variable, 6.
- 6x/6 = 36/6
=> x = 6
Types of Algebraic Equations
Algebraic equations are of many types. A few of the equations in algebra are:
All the polynomial equations are a part of algebraic equations like the linear equations. A polynomial equation comprises variables, exponents & coefficients.
- Linear equations: a+bx=c (a not equal to 0)
A quadratic equation is a polynomial equation of degree/power 2 in one variable of type f(x) = ax2 + bx + c.
- Quadratic Equations: ax2+bx+c=0 (a not equal to 0)
The cubic polynomials are polynomials with degree/power 3. All the cubic polynomials are also algebraic equations.
- Cubic Polynomials: ax3+bx2+cx+d=0
Rational Polynomial Equations
All the trigonometric equations are all considered as algebraic functions. For a trigonometry equation, the expression includes the trigonometric functions of a variable.
- Trigonometric Equations: cos2x = 1+4sinx