The set of ordered pairs represents a linear relationship is Table-II.
What are linear functions?
A mathematical function in which the variables appear only in the first degree, are multiplies by constants, and are combined only by addition and substraction.
Now from the given table we can take values of x and y,
So from Table-I ,
at x₁ = 0 ⇒ y₁ = 1
at x₂ = 1 ⇒ y₂ = 2
at x₃ = 2 ⇒ y₃ = 4
at x₄ = 3 ⇒ y₄ = 8
at x₅ = 4 ⇒ y₅ = 16
Hence,
x₂- x₁ = 1 ; y₂-y₁ = 1
x₃- x₂ = 1 ; y₃-y₂= 2
x₄-x₃ = 1 ; y₄-y₃ = 4
x₅-x₄ = 1 ; y₅-y₄ = 8
Here we can see that the for values of y there is no constant rate of change.
Hence,this ordered pair does not represents linear relationship.
Similarly, for Table-II
at x₁ = -4 ⇒ y₁ = 6
at x₂ = -1 ⇒ y₂ = 4
at x₃ = 2 ⇒ y₃ = 2
at x₄ = 5 ⇒ y₄ = 0
at x₅ = 8 ⇒ y₅ = -2
Hence,
x₂- x₁ = 3 ; y₂-y₁ = -2
x₃- x₂ = 3 ; y₃-y₂= -2
x₄-x₃ = 3 ; y₄-y₃ = -2
x₅-x₄ = 3 ; y₅-y₄ = -2
Here we can see that the for values of x and y there is constant rate of change.
Hence,this ordered pair represents linear relationship.
Similarly for Table-III
at x₁ = 0 ⇒ y₁ = 0
at x₂ = 1 ⇒ y₂ = 1
at x₃ = 2 ⇒ y₃ = 4
at x₄ = 3 ⇒ y₄ = 9
at x₅ = 4 ⇒ y₅ = 16
Hence,
x₂- x₁ = 1 ; y₂-y₁ = 1
x₃- x₂ = 1 ; y₃-y₂= 3
x₄-x₃ = 1 ; y₄-y₃ = 5
x₅-x₄ = 1 ; y₅-y₄ = 7
Here we can see that the for values of y there is no constant rate of change.
Hence,this ordered pair does not represents linear relationship.
Similarly for Table-IV
at x₁ = -2 ⇒ y₁ = 0
at x₂ = -2⇒ y₂ = 1
at x₃ = 2 ⇒ y₃ = 2
at x₄ = 7 ⇒ y₄ = 3
at x₅ = 10 ⇒ y₅ = 4
Hence,
x₂- x₁ = 0 ; y₂-y₁ = 1
x₃- x₂ = 4 ; y₃-y₂= 1
x₄-x₃ = 5 ; y₄-y₃ = 1
x₅-x₄ = 3 ; y₅-y₄ = 1
Here we can see that the for values of x there is no constant rate of change.
Hence,this ordered pair does not represents linear relationship.
Hence,The set of ordered pairs represents a linear relationship is Table-II.
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