We have learned about equations in the earlier classes. An equation is a statement of equality of two expressions. The two sides of the equality sign are referred to as the left-hand side (LHS) and the right-hand side (RHS) of the equation.
For example, in the equation 3x + 4 = 8, where 3, 4, and 8 are the constants, and x is the variable. The LHS is given by the expression 3x + 4 and the RHS is given by the constant 8. The equation remains unchanged if we carry out the same operation on both sides of the equation.
To solve an equation, we carry out a series of identical Mathematical operations on two sides of the equation such that the unknown variable is one side and its value is obtained on the other side.
Equation: An equation is a statement of equality of two algebraic expressions involving constants and variables.
Based on the degree and variable in the equations, they are classified as linear and nonlinear equations.
An equation in which the maximum degree of a term is one is called a linear equation. Or we can say that a linear equation that has only one variable is called a linear equation in one variable.
A linear equation values when plotted on the graph forms a straight line.
The general form of a linear equation is ax + b = c, where a, b, c are constants and a0 and x and y are variable.
For Example: x + 7 = 12, 5/2x - 9 = 1, x2 + 1 = 5 and x/3 + 5 = x/2 - 3 are equation in one variable x.
Here the highest power of each equation is one.
2x + 3y = 15, 7x - y/3 = 3 are equations in two variables x and y.
When the linear equation is plotted on the graph we get the below figure.
An equation in which the maximum degree of a term is 2 or more than two is called nonlinear equations.
For example 3x2 + 2x + 1 = 0, 3x + 4y = 5, this are the example of nonlinear equations, because equation 1 have highest degree of 2 and second equation have variable x and y.
The nonlinear equation values when plotted on the graph forms a curve.
The general form of a nonlinear equation is ax2 + by2 = c, where a, b, c are constants and a0 and x and y are variables.
When plotted on the graph we get the below curve
Understanding the difference between linear and nonlinear equations is foremost important. Here is the table which will clarify the difference between linear and nonlinear equations. So let us understand what are linear and nonlinear equations exactly.
Let us understand what are linear and nonlinear equations with the help of some examples.
Example1: Solve the Linear equation 9(x + 1) = 2(3x + 8)
9(x + 1) = 2(3x + 8)
Expand each side
9x + 9 = 6x + 16
Subtract 6x from both the sides
9x + 9 - 6x = 6x + 16 - 6x
3x + 9 = 16
Substract 9 from both the sides
3x + 9 - 9 = 16 - 9
3x = 7
Divide each by 3
3x/3 = 7/3
Example 2 : Solve the nonlinear equation
3x2 - 5x + 2 = 0
3x2 - 5x + 2 = 0
3x2 - 3x - 2x + 2 = 0
3x(x - 1) - 2(x - 1) = 0
( 3x - 2)( x - 1) = 0
(3x - 2) = 0 or (x - 1) = 0
x = 2/3 or x = 1
Q. Solve the following linear equation and find the value of x
3(5x + 6) = 3x - 2
(2x +9)/5 = 5
Q. Solve the nonlinear equations
7x2 = 8 - 10x
3(x2 - 4) = 5x
How do I know that an equation is a linear or nonlinear equation?
To determine whether the given equation is linear we have to determine that a given equation is in the format
y = mx + c
where m is the slope
x and y are the variables
c is the y-intercept.
For example y = 2x + 1, here the equation has the highest degree as one So it is a linear equation.
A nonlinear equation will not match this equation.
You can also test an equation is linear or nonlinear by plotting it on the graph.
If an equation gives a straight line then that equation is a linear equation.
Example: y = 2x + 1 is the equation can be represented on the graph as
Here it represents a straight line so it is a linear equation.
How to solve a linear equation?
To solve a linear equation we use the idea of a balance to find the value of x. We have to keep both the right-hand side and left-hand side balance. We can maintain this status by performing the same operation by on both sides, such as adding subtracting, multiplying, or dividing by the same numbers.
Here are the following steps to solve a linear equation:
Step 1: Start by moving all of the terms that contain a variable to the left-hand side of the equation.
Step 2:Move the terms that do not contain variables to the right-hand side of the equation.
Step 3: Look at the variable and determine if there are any other operations being performed on it.you will get the value.
Step 4: Check your answer for accuracy. To do this, put the value back into the original equation.