Informative line

Relationship Between Variables

Relationship between Two Number Patterns

  • A relation can be found between two patterns.
  • There should be two patterns with a specific rule.

Consider the following example:

Generate two number patterns starting from zero using the given rules. Then compare and find the relationship between the two patterns.

(i) Add \(4\) 

(ii) Add \(6\)

(i) There are two patterns as follows:

     Pattern \((1)\) - Addition of \(4\)

     Pattern \((2)\) - Addition of \(6\)

(ii) Starting from Zero,

     \(0, \;4, \;8, \;12, \;16, \;20,...\) [Pattern \((1)\) series]

     \(0, \;6, \;12, \;18, \;24, \;30,...\)[Pattern \((2)\) series]

(iii) Write both number patterns in ordered pairs with corresponding terms.

\((0, 0), \;(4, 6), \;(8, 12), \;(12, 18), \;(16, 24), \;(20, 30),.....\)

(iv) Try to find the relation between corresponding terms. Compare each pair, there are differences of \(0,\;2,\;4,\;6,\;8,\;10,....\)

This relation can be written as \(2n\)

 where n = Whole number

Illustration Questions

Generate two number patterns starting from 2 using the given rules: (i) Multiply \(2\) (ii) Add \(4\)  Compare both the patterns and find the difference between the \(5^{th}\) terms of both the patterns. 

A \(2\)

B \(4\)

C \(10\)

D \(14\)

×

Generate two number patterns starting from 2 using the given rules: (i) Multiply \(2\) (ii) Add \(4\)  Compare both the patterns and find the difference between the \(5^{th}\) terms of both the patterns. 

A

\(2\)

.

B

\(4\)

C

\(10\)

D

\(14\)

Option D is Correct

Tables

  • The relationship between dependent and independent variables can be represented with the help of a table.
Independent Dependent
Input Output
  • A table that shows how a value changes according to a rule is known as input-output table.
  • The independent variable is always in the left handside column, while the dependent variable is in the right handside column.
  • The value of the output depends on

(i) the value of the input.

(ii) the rule.

  • The rule shows the relationship between input and output.
  • Tables can be used to represent an equation.

Consider an example: Make a table from the given equation: \(2x+1=y\)

  • To make a table from the given rule, give inputs in the equation and get outputs.

\(2x+1=y\)

Put \(x=1\), then \(2(1)+1=y\)

\(\Rightarrow\;2+1=y\)

\(\Rightarrow\;3=y\)

Put \(x=2\), then \(2(2)+1=y\)

\(\Rightarrow4+1=y\)

\(\Rightarrow\;5=y\)

Put \(x=3\), then \(2(3)+1=y\)

\(\Rightarrow\;6+1=y\)

\(\Rightarrow\;7=y\)

Put \(x=4\), then \(2(4)+1=y\)

\(\Rightarrow\;8+1=y\)

\(\Rightarrow\;9=y\)

  • Write inputs and outputs in ordered pairs in the form: (input, output)

\((1,\;3),\;(2,\;5),\;(3,\;7),\;(4,\;9)\)

  • Now, write inputs on the left side and outputs on the right side.
Input \((x)\) Output \((y)\)
\(1\) \(3\)
\(2\) \(5\)
\(3\) \(7\)
\(4\) \(9\)

Illustration Questions

An equation is given: \(3x+4=y\) Which option correctly represents the input-output table for the given equation?

A   \(x\) \(y\) 1 3 2 4 3 5

B     \(x\) \(y\) 1 7 2 8 3 9  

C   \(x\) \(y\) 1 7 2 10 3 13  

D   \(x\) \(y\) 1 1 2 2 3 3  

×

The given equation is

\(3x+4=y\)

(i) Give inputs in the equation and get outputs.

Put \(x=1\), then \(3(1)+4=y\)

\(\Rightarrow\;3+4=y\)

\(\Rightarrow\;7=y\)

Put \(x=2\), then \(3(2)+4=y\)

\(\Rightarrow\;6+4=y\)

\(\Rightarrow\;10=y\)

Put \(x=3\), then \(3(3)+4=y\)

\(\Rightarrow\;9+4=y\)

\(\Rightarrow\;13=y\)

(ii) Write inputs and outputs in ordered pairs in the form: (input, output).

\((1,\;7)\)

\((2,\;10)\)

\((3,\;13)\)

Now, make a table from the ordered pairs.

 

\(x\) \(y\)
1 7
2 10
3 13

Hence, option (C) is correct.

An equation is given: \(3x+4=y\) Which option correctly represents the input-output table for the given equation?

A

 

\(x\) \(y\)
1 3
2 4
3 5
.

B

 

 

\(x\) \(y\)
1 7
2 8
3 9

 

C

 

\(x\) \(y\)
1 7
2 10
3 13

 

D

 

\(x\) \(y\)
1 1
2 2
3 3

 

Option C is Correct

Completion of Input-Output Table

  • The relationship between dependent and independent variables can be represented with the help of a table.
Independent Dependent
Input Output

 

  • A table that shows how a value changes according to a rule is known as the input-output table.
  • The independent variable is always in the left hand side column, while the dependent variable is in the right hand side column.
  • The value of the output depends on

(i) the value of the input.

(ii) the rule.

  • The rule shows the relationship between input and output.
  • Tables can be used to represent an equation.

Consider an example: Make a table from the given equation: \(2x+1=y\)

  • To make a table from the given rule, give inputs in the equation and get outputs.

\(2x+1=y\)

Put \(x=1\), then \(2(1)+1=y\)

\(\Rightarrow\;2+1=y\)

\(\Rightarrow\;3=y\)

Put \(x=2\), then \(2(2)+1=y\)

\(\Rightarrow4+1=y\)

\(\Rightarrow\;5=y\)

Put \(x=3\), then \(2(3)+1=y\)

\(\Rightarrow\;6+1=y\)

\(\Rightarrow\;7=y\)

Put \(x=4\), then \(2(4)+1=y\)

\(\Rightarrow\;8+1=y\)

\(\Rightarrow\;9=y\)

  • Write inputs and outputs in ordered pairs in the form: (input, output)

\((1,\;3),\;(2,\;5),\;(3,\;7),\;(4,\;9)\)

  • Now, write inputs on the left side and outputs on the right side.
Input\((x)\) Output \((y)\)
\(1\) \(3\)
\(2\) \(5\)
\(3\) \(7\)
\(4\) \(9\)

Illustration Questions

Jacob has \(3\) times less than \(10\) candies as much as Marc has. They plot a table as shown. Which one of the following options represents the correctly completed table?          Number of candies \(x\;\text{(Marc)}\) \(y\; \text {(Jacob)}\)           \(0\)            \(10\)           \(1\)             \(2\)              \(4\)           \(3\)  

A \(\begin{array} {|c|c|} \hline x&y\\ \hline 0&10\\ \hline 1&\color{red}6\\ \hline 2&4\\ \hline 3&\color{red}1\\ \hline \end{array}\)  

B \(\begin{array} {|c|c|}\hline x&y\\ \hline 0&10\\ \hline 1&\color{red}3\\ \hline 2&4\\ \hline 3&\color{red}5\\ \hline \end{array}\)  

C \(\begin{array} {|c|c|}\hline x&y\\ \hline 0&10\\ \hline 1&\color{red}6\\ \hline 2&4\\ \hline 3&\color{red}2\\ \hline \end{array}\)  

D \(\begin{array} {|c|c|} \hline x&y\\ \hline 0&10\\ \hline 1&\color{red}7\\ \hline 2&4\\ \hline 3&\color {red}1\\ \hline \end{array}\)

×

Jacob has \(3\) times less than \(10\) candies as much as Marc has. They plot a table as shown. Which one of the following options represents the correctly completed table?          Number of candies \(x\;\text{(Marc)}\) \(y\; \text {(Jacob)}\)           \(0\)            \(10\)           \(1\)             \(2\)              \(4\)           \(3\)  

A

\(\begin{array} {|c|c|} \hline x&y\\ \hline 0&10\\ \hline 1&\color{red}6\\ \hline 2&4\\ \hline 3&\color{red}1\\ \hline \end{array}\)

 

.

B

\(\begin{array} {|c|c|}\hline x&y\\ \hline 0&10\\ \hline 1&\color{red}3\\ \hline 2&4\\ \hline 3&\color{red}5\\ \hline \end{array}\)

 

C

\(\begin{array} {|c|c|}\hline x&y\\ \hline 0&10\\ \hline 1&\color{red}6\\ \hline 2&4\\ \hline 3&\color{red}2\\ \hline \end{array}\)

 

D

\(\begin{array} {|c|c|} \hline x&y\\ \hline 0&10\\ \hline 1&\color{red}7\\ \hline 2&4\\ \hline 3&\color {red}1\\ \hline \end{array}\)

Option D is Correct

Linear or Non-linear Graphs

  • A graph can be linear or non-linear.
  • When the solution of an equation on a graph forms a straight line, it is called a linear graph.

For example: Consider the graph shown.

Here, the solution of an equation forms a straight line so, it is a linear graph.

When the solution of an equation on a graph does not form a straight line, it is called a non-linear graph.

For example: Consider the graph shown.

Here, the solution of an equation forms a zig-zag line so, it is a non-linear graph.

Steps to identify linear or non-linear relationship:

(i) Plot the coordinates on the graph.

(ii) Connect the points through lines.

If the solution makes a straight line, it is a linear relationship and if it does not, it is a non-linear relationship.

  • Relationship represents the relation between \(x\) and \(y\), and shows how dependent variable changes according to the independent variable.

Illustration Questions

Which one of the following graphs represents the linear relationship?

A

B

C

D

×

Which one of the following graphs represents the linear relationship?

A image
B image
C image
D image

Option D is Correct

Real Life Examples to Represent through Table

  • Equations and tables can also be used to represent real life information, mathematically.

Consider an example:

Taylor is going for horse riding.

The riding charge is \($7/hr\).

  • Let \(h\) represents the number of hours of riding and \(c\) represents the total cost.

Equation is \(c=7h\)

To make a table from the equation, give inputs to get outputs.

If \(h=1\), then \(c=7×1\)

\(\Rightarrow\;c=7\)

If \(h=2\), then \(c=7×2\)

\(\Rightarrow\;c=14\)

If \(h=3\), then \(c=7×3\)

\(\Rightarrow\;c=21\)

If \(h=4\), then \(c=7×4\)

\(\Rightarrow\;28\)

Write in ordered pairs: \((1,7),\,(2,14),\,(3,21),\,(4,28) \)

 

Hours (h) Cost (c)
1 7
2 14
3 21
4 28

Illustration Questions

Emma's age is twice the age of Cody. Which one correctly represents the relationship between Emma and Cody's ages?

A     Cody (\(C\)) Emma (\(E\)) 10 20 11 22 12 24 13 26  

B   Cody (\(C\)) Emma (\(E\)) 10 11 11 12 12 13 13 14  

C   Cody (\(C\)) Emma (\(E\)) 10 12 11 13 12 14 13 15  

D   Cody (\(C\)) Emma (\(E\)) 10 13 11 15 12 30 13 25  

×

Emma's age is twice the age of Cody. Which one correctly represents the relationship between Emma and Cody's ages?

A

 

 

Cody

(\(C\))

Emma

(\(E\))

10 20
11 22
12 24
13 26

 

.

B

 

Cody

(\(C\))

Emma

(\(E\))

10 11
11 12
12 13
13 14

 

C

 

Cody

(\(C\))

Emma

(\(E\))

10 12
11 13
12 14
13 15

 

D

 

Cody

(\(C\))

Emma

(\(E\))

10 13
11 15
12 30
13 25

 

Option A is Correct

Graphing Equations

  • Equations and tables can also be graphed to represent a real-life situation.
  • We can represent an equation by graphical method.
  • If we have an equation in terms of two variables and there is a dependent and an independent variable, then we can plot a graph between them.
  • A graph shows the relationship between two quantities in the form of increment and decrement.

When graphing:

(i) The independent variable is always across the \(x-\)axis.

(ii) The dependent variable is always up the \(y-\)axis.

Steps for graphing equations:

(i) Write an equation to represent the situation.

(ii) Make an input-output table from the equation.

(iii) Once we have the equation, we can write coordinates in the form: (independent variable, dependent variable)

Coordinates: A pair of values that shows an exact position on a coordinate plane.

(iv) Plot the coordinates of the equation on the graph.

For example: A bakery shopkeeper makes cookies. The table shows the relationship between the time in hours \((x)\) and the number of cookies the shopkeeper makes \((y)\).

  • We can write an equation from the given table.

\(\begin{array} {|c|c|} \hline (x)&(y)\\ \text{(In hours)}&\text{(Number of cookies)}\\ \hline 1&20\\ \hline 2&40\\ \hline 3&60\\ \hline 4&80\\ \hline \end{array}\)

  • The equation for the situation is, \(y=20x\)
  • Thus, \(20\) cookies are being made each hour.

  • The value of \(y\) depends upon the value of \(x.\)

Hence, \(y\) (the number of cookies) is the dependent variable.

\(x\) (the time in hours) is the independent variable.

  • Now, write coordinates from the given table.

\(\begin{array} {|c|c|} \hline (x)&(y)&\text{Coordinates}\\ &&(x,\,y)\\ \hline 1&20&(1,\;20)\\ \hline 2&40&(2,\;40)\\ \hline 3&60&(3,\;60)\\ \hline 4&80&(4,\;80)\\ \hline \end{array}\)

 

  • Plot the coordinates on the graph from the table.
  • From this plotted graph, we can observe the relation between \(x\) and \(y.\)
  • If one value increases, the other value also increases.

Illustration Questions

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A

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Notice (8): Undefined offset: 0 [APP/View/Elements/theorypage.ctp, line 115]

B

Notice (8): Undefined offset: 1 [APP/View/Elements/theorypage.ctp, line 128]
Notice (8): Undefined offset: 1 [APP/View/Elements/theorypage.ctp, line 132]

C

Notice (8): Undefined offset: 2 [APP/View/Elements/theorypage.ctp, line 145]
Notice (8): Undefined offset: 2 [APP/View/Elements/theorypage.ctp, line 148]

D

Notice (8): Undefined offset: 3 [APP/View/Elements/theorypage.ctp, line 161]
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×

Notice (8): Undefined index: Question [APP/View/Elements/theorypage.ctp, line 222]

Notice (8): Undefined index: Question [APP/View/Elements/theorypage.ctp, line 224]
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Option A is Correct

Writing an Equation for a Given Graph

  • A graph shows the relationship between two variables.
  • This relation can be defined through an equation mathematically.
  • Relation could be found by observing the curve of the graph, whether increasing or decreasing.

To write an equation from a given graph, following steps should be followed:

(i) Write the coordinates from the given graph.

(ii) Try to observe the relation between the coordinate pairs.

(iii) Write the relation in an equation form.

For example:

The following graph shows the relationship between the number of miles traveled and the number of hours.

  • Write coordinates \((x,\;y)\) from the given graph:

\((2,\;10)\\ (4,\;20)\\ (6,\;30)\)

 

  • Try to find the relation between the pairs.

The first coordinate is, \((2,\;10)\)

Here, \(2\) is multiplied with \(5\) to get \(10\).

The second coordinate is, \((4,\;20)\)

\(4\) is multiplied with \(5\) to get \(20\).

The third coordinate is \((6,\;30)\)

\(6\) is multiplied with \(5\) to get \(30\).

So, we can write a rule: \(y=5x\)

  • We will check to be sure that our rule is working or not.

\(y=5x\)

Put \(x=1\), then \(y=5×1=5\)

Put \(x=2\), then  \(y=5×2=10\)

Put \(x=3\), then \(y=5×3=15\) and so on.

It means our rule is working.

  • So, \(y=5x\) is the required equation.
  • After finding the rule for the graph, we can get next coordinates.
  • We can use trial and error method also, to find the rule for the graph from the given options.

Illustration Questions

Which one of the following rule is correct for the given graph?

A \(y=10+x\)

B \(y=2x\)

C \(y=x+20\)

D \(y=20x\)

×

Which one of the following rule is correct for the given graph?

image
A

\(y=10+x\)

.

B

\(y=2x\)

C

\(y=x+20\)

D

\(y=20x\)

Option D is Correct

Practice Now