Applications of Linear Graph

In today's life, we see many changes in which the value of distant quantities depend on changes in the values of other quantities. For example,if the number of families visiting a restaurant increases, the income of the restaurant increases and, vice versa, if the Indian population is employed or unemployed, the time taken to finish a task decreases or increases. Hence, in some situations, the value of one quantity increases with the decrease in the value of another quantity whereas in some situations the value of one quantity decreases with an increase in the value of another quantity. Hence, two quantities are either included in direct proportion or indirect proportion. The relationship between these two quantities or the applications of linear graphs can be described in an arithmetic or graphical way (using graphs). Sometimes these two quantities show a linear dependence, in other words, changes in the value of one quantity is proportional to the first power of variation in the value of another quantity. We usually describe these linear graph applications through linear graphs.

In this article, we will discuss linear graph applications, real life applications of straight line graph, linear application and graph etc. 

What is a Linear Graph

The linear graph is a straight graph or straight line which is drawn on a plane and intersecting points on x and y coordinates. Linear equations are used in everyday life and a straight line is formed graphing those relations in a plane. These linear graph applications are described through linear graphs.

Linear Graph Equation

As we know, the linear graph form a straight line and represent the following equation

y = ax+b

In the above equation, 'a' represents the gradient of the graph and 'b' in the graph represents y-intercept.

The gradient between any two points (x₁, y₁) and (x₂, y₂) are any two points drawn on the linear or straight line.

The value of gradient a is defined as the ratio of the difference between the  y-coordinates and the x-coordinates.

a = (y₂-y₁) /(x₂ - x₁)

Or y-y₁= a (x – x₁)

The linear equation can also be written as,

Ax + by + c= 0

Linear Applications and Graph

The given situation  below shows linear applications and graphs to describe the situation.

Sonia can drive a two-wheeler continuously at a speed of 20 km/hour. Construct a distance-time graph for this situation. Through the linear graph, calculate

The time taken by Sonia to ride 100 km.

The total distance covered by Sonia in 3 hours.

Solution: Now, we will construct a graph with the help of the above values given in the table.

Let us now construct a table that represents distance traveled and time taken by Sonia to cover the total distance of 100 km in the particular time interval.

Let us now draw a graph with the help of the values given above in the tabulated form.

(image will be uploaded soon)

Through the above linear application and graph, we can calculate the required values,

Time taken by Sonia to cover 100 km

In the above linear graph 

X-coordinate of the graph corresponding to the Y-coordinate of the graph at 100 = 5.

Hence, the time taken by Sonia to cover 100 km is 5 hours.

The total distance covered by Sonia in 3 hours

In the above linear graph,

Y-coordinate of the graph corresponding to the X-coordinate of the graph at 3 = 60.

Hence, the total distance covered by Sonia in 3 hours = 60 km.

Real-Life Applications of Straight-Line Graph

Some of the real-life applications of the straight line graph are given below:

Future contract markets and opportunities can be described through straight line graphs.

Straight line graph used in medicine and pharmacy to figure out the accurate strength of drugs.

Straight line graphs are used in the research process and the preparation of the government budget.

Straight line graphs are used in Chemistry and Biology.

Straight line graphs are used to estimate whether our body weight is appropriate according to our height.

Solved Examples

1. Plot the below coordinates on the graph and answer the following questions.

(2, 210), (5, 420), (7, 560), (6, 490), (3, 280), (1, 140), (8, 630)

1. Does the graph drawn from the above coordinates represent a linear graph?

2. Find the value of x-coordinate if y-coordinate corresponds to 350?

3. Calculate the value of y-coordinate for which x-coordinate is 11, If the graph drawn is linear.

Solution: 

The graph for the above coordinates is shown below

(image will be uploaded soon)

The graph drawn is linear.

The x and y coordinates will be (4,350).

The value of y coordinate or which x-coordinate is 11 will be 840. The x and y coordinates will be (11, 840).

2. A security job pays 20 USD per hour. A graph of income depending on hours worked is drawn  below. With the help of the graph shown below, determine how many hours are needed to earn $60.

(image will be uploaded soon)

Solution: By determining $60 amount on the vertical axis which represents income in dollars in the graph, you can follow a horizontal line ( represent time in hour) through the value, until it meets the graph. Follow a vertical line straight down from there, until it meets the horizontal axis. Hence, the value in hours is 3. It implies that the 3 hours of work is needed to earn $60.

Quiz Time

1. The important attributes in the graphical representation of the straight line are

X-intercept

Y-intercept

a- intercept

Both a and b

2. What will be the y-intercept of the ordered pair (8,6) of the linear equation?

-6

6

-8

8

Introduction to Linear Graphs

A linear graph is described as the graphical representation of a straight line. It is the type of graph that is represented in the form of a straight line. Linear graphs are basically used to show a relationship between two or more quantities. If the graph is represented in a single straight line then it is known as a linear graph. So linear means a straight line. The linear graph is a straight line graph and this graph comprises two axes known as x-axis and y-axis.

• The horizontal axis is the x-axis.

• The vertical axis is the y-axis.

Linear Equations

A linear equation is the type of equation consisting of the highest power of the variable as 1. Linear equations are also known as one-degree equations. The linear equation in one variable has its standard form as : 

Ax + B = 0. 

Here, x = variable, A and B = constants. 

The linear equation in two variables has its standard form as : 

Ax + By = C. 

Here, x and y = variables, A, B and C = constants.

Solving and Graphing Linear Equations

Solving linear equations will produce a straight line in a graph. The linear equation’s general formula is y = mx + b, 

Here, m = slope of the line, 

b = point on the line that crosses the y-axis. 

Now, on graph paper draw a table of values by putting x values into the equation. Only two points on the graph are enough to draw a line representing the linear equation. 

For example, if the line is y = 2x then two points are : y = 2(1) = 2, providing (1,2) as a coordinate and y = 2(10) =20, providing (10,20) as a coordinate.

Now with the identified values draw an X-Y axis on graph paper. The X-Y axis looks like a cross sign. The center of this axis is the center of a graph paper and labels it as 0. For labelling the X-axis, mark 10 squares to the left of the origin and move to the right, labelling each square with a number from -10 to 10. For labelling the Y-axis, mark 10 squares above the origin and move down, labelling each square with a number from -10 to 10. Coordinated points are now graphed. The coordinate point of values (1,10) represents (x,y) on the graph that means 1 on the x-axis then tracing upward and marking y = 10 and labelling this point as (1,10). Now the same process for labelling (10,20). After marking the points connect those two coordinate points in a straight line with a ruler. The graph generated is a linear graph. This graph can also be used to solve the equation for any value.

Applications of linear graphs

1. Mohit can ride a bike constantly at a speed of 20 km/hour. Draw a distance-time graph for this situation. With the help of a linear graph, calculate

The time taken by Mohit to ride 100 km.

The distance covered by Mohit is 3 hours.

Ans: Draw a table of values correlating the time of travel and the distance traveled by Mohit in the particular time interval.

Now draw a graph with the tabulated values,

(Image will be uploaded soon)

By watching the graph the expected values can be calculated.

So, time is taken to cover 100 km by Mohit :

From the graph, the X-coordinate of the graph corresponds to the Y-coordinate of the graph at 100 = 5. 

Hence, the time taken to cover 100 km by Mohit = 5 hours. Distance covered by Mohit in 3 hours :

From the graph, the Y-coordinate of the graph corresponding to the X-coordinate of the graph at 3 = 60

Hence, the distance covered by Mohit in 3 hours =  60 km.

Real-life application of Linear Graph

Real-life applications of linear graphs are :

• Linear graphs are important for veterinarians to administer the correct dosage of medicine.

• Economics data on House Prices follows a linear graph to identify the increases and decreases.

• Market Share of Products is constantly monitored by gaming console companies with the help of linear graphs.

• Linear graphs are used to analyze and predict future markets and opportunities.

• Linear graphs are used in Biology and Chemistry. Linear Graphs are used in medicine and pharmacy to work out the correct strength of drugs.

• Linear Graphs are used in the analysis and preparation of Government Budgets.

Solved examples

1. Mrs Mary asks John to identify whether the given equation 3x - 7y = 16 forms a linear graph or not without plotting its values. Now help John to figure out whether it is a linear graph or not.

Ans: The equation 3x - 7y = 16 is a type of linear equation in two variables. John first needs to identify the type of equation. Next, John needs to remember that any linear equation in two variables always represents a straight line. The above two points are enough to know about the nature of the graph. Therefore, the given equation represents a linear graph.

2. Mikel has to prepare a linear graph for the equation 2x + y = 8. Complete the table below for the above equation.

Ans: 

Case 1 : y = 8, x = ?

2x + y = 8

2x + 8 = 8

2x = 8−8 = 0

x = 0

Case 2 : x = 4, y = ?

2x + y = 8

2 × 4 + y = 8

8 + y = 8

y = 0

Case 3 : x =−2,y = ?

2x + y = 8

2 × (−2) + y = 8

−4 + y = 8

y = 8 + 4 =12

The completed solutions are: