Solved Examples and Worksheet for Classifying Relations as Linear or Nonlinear

Q1Which of the input-output tables forms a linear function?


A. Table 1
B. Table 2
C. Table 3
D. Table 4
E. Table 3 and Table 4

Solutions
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Step: 1
A table is said to form a linear function if the ratio of change in input to output is same through out the table.
Step: 2
In the first table Change in f(x)Change in x is not same through out. So, it is ruled out.
Step: 3

Step: 4
The change in input (x) is 3 and the change in output f(x) is 4.
Step: 5
Change in f(x)Change in x = 43
Step: 6
In the third and fourth tables Change in f(x)Change in x is not same through out. So, they are ruled out.
Step: 7
Therefore,among the choices, table-2 forms a linear function.
Correct Answer is :   Table 2
Q2Identify a table that represents a non-linear function.


A. Table 2
B. Table 3
C. Table 5
D. Table 1
E. Table 4

Solutions
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Step: 1
A table is said to form a linear function if the ratio of change in output to input is same throughout the table.
Step: 2
In the tables 1, 3, 4, and 5, change in f(x)change in x is same throughout.
Step: 3
So, tables 1, 3, 4, and 5 represents linear functions except table 2.
Step: 4
So, table 2 represents a non-linear function.
Correct Answer is :   Table 2
Q3Which of the following equations is a linear equation?

A. 2 + 3x = 4
B. x3 - 6 = 1
C. (x + 2)2 = 6
D. 3x - 5 = 8x2

Solutions
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Step: 1
An algebraic equation is said to be linear if the variable or variables in the equation are of first degree.
Step: 2
Among the choices, the equation 2 + 3x = 4 is a linear equation.
Correct Answer is :   2 + 3x = 4
Q4Identify a table that represents a linear function.


A. Table 2
B. Table 3
C. Table 1
D. Table 4

Solutions
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Step: 1
A table is said to form a non-linear function if the ratio of change in output to input is same throughout the table.
Step: 2
In the tables 1, 2 and 4, change in f(x)change in x is not same throughout.
Step: 3
  [Consider table 3.]

Step: 4
The change in input (x) is 1 and the change in output f(x) is 2.
Step: 5
change in f(x)change in x = 21
Step: 6
Among the choices, Table 3 forms a linear function.
  [The ratio of the change in f(x) to the change in x is same throughout the table.]
Correct Answer is :   Table 3
Q5Which among the following is a linear function?
(i) 4x + y - 6 = 0
(ii) 2y - (12x) = 1
(iii) xy - 3 = 0

A. (i), (ii), and (iii)
B. (i) and (ii)
C. only (i)
D. only (ii)

Solutions
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Step: 1
A linear function is of the form y = mx + b.
Step: 2
4x + y - 6 = 0 gives y = 6 - 4x
  [Keep 'y' on one side, bring the rest to the other side.]
Step: 3
2y - (12x) = 1 gives y = (6x) + 12
Step: 4
xy - 3 = 0 gives y = 3x
Step: 5
Among the given equations, only 4x + y - 6 = 0 represents a linear function.
Correct Answer is :   only (i)
Q6Which of the following equations is not a linear equation?
A. x = 42
B. 3x2 = 15
C. 5x - 7y = 16
D. y = 5x - 3

Solutions
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Step: 1
A function in the form of f(x) = mx + c is called a linear function.
  [In a linear function, the exponent of x must be 1.]
Step: 2
Among the choices 3x2 = 15 is not in the form of a linear function.
  [The exponent of x is 2.]
Step: 3
So, the equation in choice B is not a linear function.
Correct Answer is :   3x2 = 15
Q7Identify the graph that represents a non-linear model.


A. Graph 3
B. Graph 2
C. Graph 1
D. Graph 4

Solutions
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Step: 1
The graph of a linear model is a straight line.
Step: 2
Graphs 1, 3, and 4 are straight lines.
Step: 3
Therefore, Graph 2 represents a non-linear model.
Correct Answer is :   Graph 2
Q8Which is a linear function?

A. x + y - 4 = 0
B. 9xy - 2 = 0
C. 5y - (19x) = 12
D. y3 = 3x

Solutions
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Step: 1
A linear function is of the form y = mx + b.
Step: 2
x + y - 4 = 0 gives y = 4 - x
  [Keep 'y' on one side, bring the rest to the other side.]
Step: 3
5y - (19x) = 12 gives y = (195x) + 125
Step: 4
9xy - 2 = 0 gives y = 29x
Step: 5
y3 = 3x is not in the form y = mx + b
Step: 6
So, x + y - 4 = 0 represents a linear function.
Correct Answer is :   x + y - 4 = 0
Q9Which of the following equations is not a linear equation?

A. 45a = 20
B. 5x - 3 = 38x
C. (2x - 7)3 = 26
D. 6 + 7x = 9x

Solutions
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Step: 1
An algebraic equation is said to be linear if the variable or variables in the equation are of first degree.
Step: 2
In the equation (2x - 7)3 = 26, x is raised to the power 3.
Step: 3
So, (2x - 7)3 = 26 is not a linear equation.
Correct Answer is :   (2x - 7)3 = 26
Q10Which of the following equations is not a linear equation?

A. x = 16
B. 7x + 25 = 18x
C. 5 + 7x = 19
D. (x - 3)2 = 9

Solutions
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Step: 1
An algebraic equation is said to be linear if the variable or variables in the equation are of first degree.
Step: 2
In the equation (x - 3)2 = 9, x is raised to the power 2.
Step: 3
So, (x - 3)2 = 9 is not a linear equation.
Correct Answer is :   (x - 3)2 = 9
Q11Which of the following equations is not a linear equation?
A. x2 - 5x - 6 = 0
B. x - 5x - 6 = 0
C. 7x = 12x + 46
D. 7x = 121

Solutions
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Step: 1
An algebraic equation is said to be linear if the variable or variables in the equation are of first degree.
Step: 2
In the equation x2 - 5x - 6 = 0, x is raised to the power 2.
Step: 3
So, x2 - 5x - 6 = 0 is not a linear equation.
Correct Answer is :   x2 - 5x - 6 = 0
Q12Which of the following situations could not be represented by a linear function?
A. Perimeter of a square as a function of side.
B. Interest for an amount of $P at 4% rate of interest for 1 year.
C. Interest for an amount of $P at r% rate of interest for 2 years.
D. Feet as a function of miles.

Solutions
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Step: 1
A linear function is function that can be written as y = mx + c.
Step: 2
Interest for an amount of $P at r% rate of interest for 2 years can be written as I = P(1 + r100)2 - P, which is not a linear function.
Step: 3
Among the choices given, 'Interest for an amount of $P at r% rate of interest for 2 years' is not a linear function.
Correct Answer is :   Interest for an amount of $P at r% rate of interest for 2 years.
Q13Which of the following equations is not a linear equation?

A. 3a + 32b = 23b + 37
B. 3x + y - 2 = - 5x + 4y + 2
C. 3x = 14
D. x4 - 2 = 23

Solutions
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Step: 1
An algebraic equation is said to be linear if the variable or variables in the equation are of first degree.
Step: 2
In the equation x4 - 2 = 23, x is raised to the power 4.
Step: 3
So, x4 - 2 = 23 is not a linear equation.
Correct Answer is :   x4 - 2 = 23
Q14Which of the following equations is in the linear form?
A. 17 - xy = x + 9
B. 5xy - yx = 6
C. - 10x - y2 + 8x = 0
D. - 5x - y = - 16

Solutions
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Step: 1
An equation of the form ax + by = c where either a ≠ 0 or b ≠ 0 is the linear equation.
Step: 2
Hence, from the given choices, '- 5x - y = - 16 ', is the linear equation.
Correct Answer is :   - 5x - y = - 16
Q15Which of the following equations is not a linear equation?
A. x = 6
B. 9 + 6x = 7
C. (x + 4)3 = 9
D. 6x - 7 = 15x

Solutions
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Step: 1
An algebraic equation is said to be linear if the variable or variables in the equation are of first degree.
Step: 2
In the equation (x + 4)3 = 9, x is raised to the power 3.
Step: 3
So, (x + 4)3 = 9 is not a linear equation.
Correct Answer is :   (x + 4)3 = 9

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