Applications of Linear Graph - Real Life Applications and Solved Examples
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
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
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.
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:
FAQs on Applications of Linear Graph
1. What is a linear graph and how is its equation typically represented?
A linear graph is a visual representation of a linear equation that always forms a perfectly straight line on a Cartesian plane. This straight line indicates a constant relationship between two variables. The most common form of its equation is the slope-intercept form, y = mx + c, where 'x' and 'y' are the variables, 'm' represents the slope (the steepness of the line), and 'c' is the y-intercept (the point where the line crosses the vertical y-axis).
2. What are some important applications of linear graphs in real life?
Linear graphs are widely used to model and understand relationships that have a constant rate of change. Some key real-life applications include:
- Business and Economics: To analyse cost, revenue, and profit. For instance, plotting the total cost (fixed cost + variable cost per unit) against the number of units produced.
- Science: To study relationships like distance-time graphs for an object moving at a constant speed, or to convert temperature scales (e.g., Celsius to Fahrenheit).
- Finance: To calculate simple interest over time, where the interest earned grows at a constant rate.
- Health and Fitness: To track calorie burn at a steady pace of exercise over a period of time.
3. How can a linear graph be used to solve a practical problem, like a mobile data plan?
A linear graph can easily solve problems involving a fixed initial cost and a variable rate. For example, consider a mobile plan with a fixed monthly rental of ₹200 that includes 5 GB of data, plus a charge of ₹20 for every extra GB used. A linear graph can be plotted with the total data used on the x-axis and the total bill amount on the y-axis. By extending the line, you can easily find out the bill for any amount of data consumed, or determine how much data you can use for a specific budget.
4. How do you plot a linear graph from an equation like 3x + y = 6?
To plot a linear graph from the equation 3x + y = 6, you need to find at least two coordinate pairs (x, y) that satisfy it. A simple method is to first rearrange the equation to y = -3x + 6. Then:
- Find the first point: Set x = 0. The equation becomes y = -3(0) + 6, so y = 6. The first coordinate is (0, 6).
- Find the second point: Set x = 1. The equation becomes y = -3(1) + 6, so y = 3. The second coordinate is (1, 3).
5. Why is a straight line graph so useful for representing relationships like speed or cost?
A straight line is useful because it visually represents a constant rate of change. In real-world scenarios like speed or cost, this is very common. For example, if a car travels at a constant speed of 60 km/h, it covers 60 km every hour. This predictability is perfectly captured by a straight line, where each increment on the time axis (x-axis) corresponds to an equal, predictable increment on the distance axis (y-axis). This makes it easy to predict outcomes and make calculations without complex formulas.
6. In a real-world linear graph, what do the 'slope' and the 'y-intercept' actually signify?
In a real-world application, the slope and y-intercept have very practical meanings:
- The slope (m) represents the rate of change. For instance, in a cost-quantity graph, the slope is the price per item. In a distance-time graph, the slope is the speed.
- The y-intercept (c) represents the initial or fixed value. It is the value of 'y' when 'x' is zero. In a taxi fare graph, it would be the fixed starting charge before the meter starts running. In a temperature conversion graph, it could be the freezing point offset.
7. When would a linear graph be an inappropriate or inaccurate model for a real-life situation?
A linear graph becomes inappropriate when the relationship between variables is not constant. It would be an inaccurate model for situations such as:
- Compound Interest: The interest earned is added to the principal, so the growth rate accelerates over time, creating a curve, not a straight line.
- Vehicle Acceleration: When a car speeds up, its velocity changes over time, so a distance-time graph would be a curve.
- Population Growth: Populations often grow exponentially, meaning the rate of growth increases as the population gets larger.
