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Linear graphs are used to represent a relationship between two or more quantities. The linear graphs are always represented in a single straight line. It includes two axes known as the x-axis and the y-axis.
| Table of Contents |
Key Terms: Linear Equation, Linear Graph, Equation, Co-ordinates, Standard Form, General equation,
What Are Linear Graphs?
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The linear graph is a graph that is in form of a straight line and represents equation-
| Y = ax+b |
Here,
- ‘a’ represents the gradient of the graph
- ‘b’ represents the y-intercept in the graph
Difference between two points (x₁, y₁) and (x₂, y₂) are any two points drawn on the linear or straight line.
The gradient ‘a’ can be defined as the ratio of the difference between the y-coordinates and the x-coordinates.
These equations can also be represented as
A = (y₂-y₁) / (x₂ - x₁)
Or
y-y₁= a (x – x₁)
This linear equation can also be expressed as
| Ax + by + c= 0 |
-
Example of Linear Graphs
The table below shows the cost of using a plumber for between 0 and 4 hours. This plumber has a $35 callout fee and charges $25 per hour
| Hours | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Cost | 35 | 35+25x1 | 35+25x2 | 35+25x3 | 35+25x4 |
It can be seen from this table that the cost function for the plumber is c = 35 + 25t, where t is the number of hours worked. This is a linear function and it has a straight-line graph
Graph Showing Time & Cost of Plumber
The graph cuts the c-axis at the point (0, 35), corresponding to the cost of the call-out fee when t = 0. The cost then increases steadily at a rate of $25 per hour.
Applications Of Linear Graphs
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The principal objective for the utilization of a linear system of linear equations is to take care of different issues utilizing two variables where one is known and the other is obscure, additionally reliant upon the first.
A portion of these uses of linear systems is:
- Geometrical problems by using two variables.
- Money or capital sums by using two variables
- A linear of problems by using two variables.
- Time distance speed rate problems by using two variables.
- Application of linear equation in business and economics
- These are used in medicine and pharmacy to figure out the accurate strength of drugs.
- Used to estimate whether our body weight is appropriate according to our height.
The Standard Form Of A Straight Line
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The equation of a line is a recipe showing the connection between the x-and y-coordinates of any point p on the line.
Assume a line has angle m and y-block (0, c), and assume that the point p is an overall point on the line with coordinates (x, y).
Also Check: Linear Equation Standard Form
The diagram above shows that the gradient of the line between the y-intercept (0, c) and the point p(x, y) is:
m = y – c / x - 0 = y - c/x
Mx = y - c
Y = mx + c
This formula equating the x- and y-coordinates of the line is the equation of the line, also called the standard form of a straight line.
Every straight line with gradient m and y-intercept (0, c) has the equation y = mx+c, and the graph of the equation y = mx+c is a straight line with gradient m and y-intercept (0, c).
The General Equation Of A Straight Line
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The gradient of a line,
m = y2 - y1
x2 - x1
is only defined when the denominator of the formula is not zero. This holds true for all lines with the only exception being the vertical lines.
Because of this, the standard form of a straight line does not apply to vertical lines.
| Example: Find the intercepts of the line 2x + 3y = 12 Solution: Put y = 0, then 2x =12Þx = 6. put x = 0, then 3y =12 Þ y = 4. the intercepts are (6, 0) and (0, 4). . . . However, to find the gradient of a line, you need to change its equation into the standard form |
Also Check: Graph of a Linear Equation in Two Variables
Things To Remember
- The important features of linear functions are inferred from their graphs.
- A linear graph is a graph that is in form of a straight line and represents an equation.
- The principal objective for the utilization of a linear system of linear equations is to take care of different issues utilizing two variables.
- Formula equating the x- and y-coordinates of the line is the equation of the line, also called the standard form of a straight line.
Sample Questions
Ques. Solve x – 3 = 0. (2 Marks)
Ans: Adding 3 to both sides of the equation
x - 3 = 0
x - 3+ 3 = 0 + 3
x = 3
Ques. Solve the simultaneous linear inequalities below graphically. (5 Marks)
a) X + y £ 2
b) Y > x - 1
c) X ³ 0 y ³ 0
Ans: Thus,
Ques. Solve the following pairs of simultaneous equations. (9 Marks)
(a) 2x + y =12
2x - y = 8
(b) x + 2y = 6
x + 5y =18
(c) 2x - y = 4
4x - y =10
Ans: 
Ques. A laboratory needs at least 300 beakers of one size and at least 400 beakers of a second size. It is decided that the total number of beakers should be less than 1200. Draw a graph showing the possible numbers of each kind of beaker. (5 Marks)
Ans: If the laboratory obtained x beakers of the first size and y beakers of the second size, then we should have
X ≥ 300, y ≥ 400, and x + y ≤ 1200
the shaded area below shows the graph of these inequalities.
Ques. What is a non-linear graph? (3 Marks)
Ans: If the plot of any function is a straight line, then it is called linear. The general equation for such a graph is Y=m.X+C. Suppose a non-accelerated (velocity (V) does not change) motion, then displacement (X) can b calculated as X=VT. Where T is the time which is similar to the equation mentioned above.
Ques. 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. (3 Marks)
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.
Ques. What will be the y-intercept of the ordered pair (8,6) of the linear equation? (1 Marks)
a) 6
b) – 6
c) 8
d) – 8
Ans: 6
Ques. Sakshi can ride a scooter constantly at a speed of 20 km/hour. Draw a distance-time graph for this situation. With the help of the linear graph, calculate the following: (5 Marks)
The time taken by Sakshi to ride is 100 km.
The distance covered by Sakshi in 3 hours.
Ans. The time of travel and the distance traveled by Sakshi in the particular time interval is
| Time (hr) | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Distance (km) | 20 | 40 | 60 | 80 | 100 | 120 |
Graph:-
- The time is taken by Sakshi to ride 100 km
From the graph,
X-coordinate of the graph corresponding to the Y-coordinate of the graph at 100 = 5
Hence, time taken to cover 100 km by Sakshi = 5 hours
- The distance covered by Sakshi in 3 hours.
From the graph,
the Y-coordinate of the graph corresponds to the X-coordinate of the graph at 3 = 60
Hence, the distance covered by Sakshi in 3 hours = 60km
Ques. Plot a graph of a linear equation y=2x+1. (2 Marks)
Ans: Calculating the value of y with respect to x
y=2x+1
y=2(-2)+1= -3 for x=-2
y=2(-1)+1= -1 for x=-1
y=2(0)+1= 1 for x=0
y=2(1)+1= 3 for x=1
y=2(2)+1= 5 forx=2
Table for co-ordinates:
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y | -3 | -1 | 0 | 3 | 5 |
Plotting the co-ordinates on graph:-
Ques. Write rach of the equation in the form of ax + by + c = 0. Indicate the values of a,b,c (5 Marks)
a) 3x + 4y = 7
b) 2X = 4y
Ans: a) 3x + 4y = 7
3x + 4y - 7 = 0
On comparing with ax + by + c = 0, we get
a = 3 , b = 4, and c = -7
b) 2x = 4y
The above equation can be written as
2x - 4y = 0
There is no constant value given in the equation,
So equation can be written as
2x - 4y + 0 = 0
On comparing with ax + by + c = 0, we get
a = 2 , b = - 4 and c = 0
Ques. The cost of one notebook is thrice the cost of a pen. Write the linear equation in two variables to represent this statement. (1 Mark)
Ans: Linear equation for above statement is:
X - 3y + 0 = 0
Ques. Write following as an equation in two variables x and y. (3 Marks)
a) X = - 5
b) 3x = 7
Ans: a) x = - 5
X + 5 = 0
x + 0 . y + 5 = 0
b) 3x = 7
3x - 7 = 0
x + 0 . y - 7 = 0
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