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An equation is defined as a formula that expresses the equality of two expressions, connecting them with equal sign (=). The expressions on the two sides of the equality sign are called the left-hand side (LHS) and right-hand side (RHS) of the equation. There are different types of equations available to solve mathematical problems. Based on the degree and variable present in the equation, they are classified into linear and non-linear equations. If the degree of the term is one then it is called a linear equation and if the degree of a term is two or more than two then it is called a nonlinear equation.
Read Also: Solving Inequalities
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Key Terms: Linear equation, Nonlinear equation, formula, Algebra, equations, General form, Maximum Degree, Straight-line
What is Linear equation?
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Linear equation is a one-degree equation which means, the highest power of the variable is always 1. It appears as straight lines when graphed in the XY plane. The standard form of linear equation with one variable is given by,
Ax + B = C
Where, x is a variable, A, B and C are real numbers and A≠B≠0.
The standard form of linear equation with two variables is given by,
Ax + By = C
Where x and y are variables, while A, B and C are real numbers.
For example, x + 2 = 5 and x/2 + 3 = 5 are linear equations with one variable x. 2x + 3y = 4 and 4x – y/2 = 15 are linear equations with two variables x and y.
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What is a Non-Linear equation?
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An equation in which the maximum degree of a variable is 2 or more than two is called a nonlinear equation; it can’t be less than 2. It forms a curved graph on the XY plane. The standard form of a nonlinear equation is given by,
Ax2 + By2 = C
Where, x and y are variables while A, B and C are constants.
For example, x2 + y3 = 4 and x2 + x + 3 =7; this are the examples of nonlinear equations because maximum degree of both the equations are 2 or more than 2.
Read More: Pair of linear equations in two variables
Difference between linear and nonlinear equations
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| Linear Equation | Non-Linear Equation |
|---|---|
| An equation in which the maximum degree of a term is one is called a linear equation. | An equation in which the maximum degree of a term is 2 or more than 2 is called a nonlinear equation. |
| Results of linear equation, when displayed on a graph, it creates a straight line. | Results of nonlinear equation, when displayed on a graph, it creates curves like hyperbola, parabola or ellipse. |
| A linear equation has a general form, y = mx + C Where m is the slope of the line, C is a constant value and x and y is the variables. | A nonlinear equation has a general form, Ax2 + By2 = C Where A, B and C are constant values and x and y are variables |
| All linear equations form a straight line; these lines can be extended to any direction but in a straight form. | Nonlinear equations form a curve and if we increase the value of the degree than the curvature of the graph increases. |
| It has a constant slope value. | It has a variable slope value. |
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Check Important Notes for Quadratic equations formula
Things to Remember
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- When the same number is added, subtracted, multiplied or divided to both sides of a linear equation than the result is unaltered.
- Unlike the nonlinear system, the output of a linear system is directly proportional to its input.
- If the differences between the outputs of the equation are consistent when you use unknown variables, then the equation is linear and if it is inconsistent then nonlinear.
- Nonlinear equations can take many shapes from simple curves to elaborate images.
- Linear equations are easier to use than nonlinear equations.
- A linear equation is used to describe the relationship between two variables, make predictions, calculate variable costs, make conversions and determine earnings.
- The nonlinear equation is used in regression analysis, logarithmic scales and S-curve graphs.
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Sample Questions
Ques.What forms a linear equation? (1 Mark)
Ans: Linear equation can be described by the form y = mx + c where, m is the slope of the line, c is a constant value and x and y is the variables. This linear equation will have a straight line passing through the origin on a graph with a constant slope value.
Ques.What is the key difference between linear and nonlinear equations? (1 Mark)
Ans: Linear equation forms a straight line on a graph, while nonlinear equation forms a curve like a parabola, hyperbola or ellipse on the graph.
Ques.Solve the linear equation 3x + 6 = 2x + 12. (1 Mark)
Ans: Given equation is 3x + 6 = 2x + 12
3x – 2x = 12 – 6
X = 6
Ques.How nonlinear equation is represented? Explain it with example. (2 Marks)
Ans: The general representation of nonlinear equation is Ax2 + By2 = C
Where, x and y are variables and A,B and C are constant values.
Example: 3x2 + 2y3 = 7
Ques.Find the value of variable x by solving the linear equation 4(x+2) = 2(2x+5). (2 Marks)
Ans: Given equation is 6(x+2) = 2(2x+8)
6x + 12 = 4x + 16
6x – 4x = 16 – 12
2x = 4
X=2
Ques.Solve the nonlinear equation 2x2- 3x +2 = 0. (2 Marks)
Ans: given equation is 2x2- 4x +2 = 0
By factorizing
2x2- 2x-2x +2 = 0
2x(x-1)- 2 (x-1) = 0
(2x-2)(x-1) = 0
2x-2 = 0 or x-1 = 0
X=1
Ques.Find the value of variables x and y from the linear equation x+4y = 2 and x=y. (2 marks)
Ans: Given equation is, x + 2y = 1 and x = y
By putting the value of x in the first equation we get,
y + 4y = 2
5y = 2
Y=2/5
∴ x = y = 2/5
Ques.Solve the given nonlinear equation 3(x2-4) = 5x to find the value of x. (2 Marks)
Ans: Given equation is, 3(x2-2) = 7x
3x2-6 = 7x
3x2 - 7x - 6 = 0
3x2- 9x +2 x - 6 = 0
3x (x-3) + 2(x-3) = 0
3x+2=0 and x-3=0
X = -2/3 or x=3
Ques.What are the features of nonlinear equations? (3 Marks)
Ans: Features of nonlinear equations are as follows:
- Nonlinear equation can provided in the simple form by Ax2 + By2 = C
- It graphed as curve on a XY plane.
- It has a variable slope range on the length of the curve therefore it provides different values at different areas of the curve.
- The maximum degree of nonlinear equation is always two or higher and if the degree increases the curvature of the graph also increases.
- Input data and output result of a nonlinear system are not directly related.
- Nonlinear equations are complex to solve
Ques.If the point (3,4) lies on the graph of the equation 3y = ax + 9, find the value of a. (3 Marks)
Ans: Given equation is 3y = ax + 9
A point lie on the graph is (3, 4)
Therefore x=3 and y=4
Now, substituting the values of x and y in the equation 3y = ax + 9,
We get, 3(4) = a(3) + 9
12 = 3a + 9
3a = 12 – 9
3a = 3
a=1
the value of a, if the point lies on the graph of the equation 3y = ax + 9 is 1.
Ques.Solve the following system of equation.
y = x2 + 6x – 8
y = 4x + 7 (3 Marks)
Ans: Given equations are y = x2 + 6x – 8, y = 4x + 7
By substituting the value of y in equation we get,
x2 + 6x – 8 = 4x + 7
x2 + 6x – 4x – 8 – 7 = 0
x2 + 2x – 15 = 0
(x – 3)(x + 5) = 0
X = -5 and x = 3
To determine the corresponding value of y for each x,
For x = -5 : y = 4 (-5) + 7 = -13…….. (-5, -13)
For x = 3 : y = 4(3) + 7 = 19……………(3, 19)
Therefore, for this system of equation we have two solutions (-5, -13) and (3, 19).
Ques.Solve the 2x + 5y = 11 and 2x – 3y = -13 and hence find the value of m for which y = mx + 2. (5 Marks)
Ans: 2x + 5y = 11………………..(i)
2x – 3y = -13……………………….(ii)
From equation (i) we get, x = (11 – 5y)/2…………………(iii)
By putting the value of x in equation (ii), we get
2[(11 – 5y)/2] – 3y = -13
11 – 5y – 3y = -13
-8y = -24
y = 3………………………………………..(iv)
By putting the value of y in equation (iii), we get
x = (11 – 15)/2 = -4/2 = -2
Therefore, x = -2 and y = 3
Also, y = mx + 2
3 = -2m + 2
-2m = 1
m = -1/2
Hence, the value of m is -1/2.
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