Graph of a Linear Equation: Definition, Notes, Examples

An equation is a statement of equality that contains one or more undetermined quantities or variables. An equation containing only a linear polynomial is called a linear equation. The value of the variable which is making the equation a real statement is called the solution or root of the equation. Graph of a linear equation is described as a linear equation represented graphically by the line whose points give the collection of solutions of the equation.

Unless all variables represent real numbers, one will be able to graph the equation by plotting sufficient points to recognise a pattern and then connect the points to include all points. In this article, we will provide detailed information on the graph of a linear equation. Continue reading to find out more!

Define Graph of a Linear Equation

We understand that each linear equation in one variable has a unique solution. Then, what about the solution of a linear equation involving two variables? If there are two variables in the equation, a solution involves a pair of values, one for \(x\) and one for \(y\) which satisfy the given equation.

Example: \(x + 2y = 0\)

When we take \(x = 0\), we get \(y = 0\), therefore \((0,\,0)\) is the solution of the equation. When we take \(x = 4\), we get \(y = – 2\), therefore \((4,\, – 2)\) is the solution of the equation. When we take \(x = 6\), we get \(y = – 3\), therefore \((6,\, – 3)\) is the solution of the equation.

So, there is no end to the various solutions of a linear equation in two variables. That is to say, a linear equation in two variables has infinitely numerous solutions.

A linear equation is represented graphically by the line whose points give the collection of solutions of the equation. This is called the graph of a linear equation.

Graph of a Linear Equation Notes

Graphical representation of linear equation in one variable:
Example: \(2x – 6 = 0\)
\( \Rightarrow 2x = 6 \Rightarrow x = \frac{6}{2} = 3\)

So, the solution of the equation is \(x = 3\). This could be represented as follows:

Graphical representation of linear equation in two variables:

For example, represent \(x + 2y = 6\) on the graph.

The solution of the linear equation \(x + 2y = 6\) can be expressed in the form of a table as follows by writing the values of \(y\) below the corresponding values of \(x\):

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Here, every point on the line satisfies the equation of the line, and every solution of the equation is a point on the line. A linear equation in two variables has been represented geometrically by a line from which the points make the collection of solutions of the equation. This is known as the graph of the linear equation. So, to get the graph of a linear equation in two variables, it is sufficient to plot two points corresponding to two solutions and join them by a line.

Note: The explanation that a degree one polynomial equation \(ax + by + c = 0\) is called a linear equation is that its geometric representation is a straight line.

Graph of Pair of Linear Equations in Two Variables

The general form for a pair of linear equations in two variables \(x\) and \(y\) is \({a_1}x + {b_1}y + {c_1} = 0\) and \({a_2}x + {b_2}y + {c_2} = 0\), where \({a_1},\,{b_1},\,{c_1},\,{a_2},\,{b_2},\,{c_2}\) are all real numbers and \(a_1^2 + b_1^2 \ne 0,\,a_2^2 + b_2^2 \ne 0\).

Conditions for the solvability of a pair of simultaneous linear equations in two variables.

A pair of linear equations that has no solution is called an inconsistent pair of linear equations. A pair of linear equations in two variables with at least one pair of solution is called a consistent pair of linear equations.  A pair of linear equations which are equivalent has infinitely many different standard solutions.  Such a pair is called a conditional pair of linear equations in two variables. Note that a conditional pair of linear equations is ever consistent.

Consistent system: A system of simultaneous linear equations is consistent if it has at least one solution.

1. When, \(\frac{{{a_1}}}{{{a_2}}} \ne \frac{{{b_1}}}{{{b_2}}}\), we get a unique solution.

2. When \(\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} = \frac{{{c_1}}}{{{c_2}}}\), there are infinitely many solutions.

In-consistent system: A system of simultaneous linear equations is said to be inconsistent if it has no solution.

1. When \(\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} \ne \frac{{{c_1}}}{{{c_2}}}\), there is no solution.

Graph of pair of linear equations in two variables:

The graphical (i.e. geometric) representation of a linear equation in two variables is a straight line such that every point on the line represents a solution of the equation, and every solution of the equation is represented by a point on the line. Let us now see what a pair of linear equations in two variables will look like graphically? A linear equation in two variables represents a straight line.

Thus, a pair of linear equations in two variables are going to be represented by two straight lines, both to be considered together. We understand that given two lines in a plane, only one of the following three options can happen:

1. The lines may intersect at a single point. If this is the case, the pair of equations has a unique solution (consistent pair of equations).
2. The lines may be parallel.  If this is the case, the equations have no solution (inconsistent pair of equations).
3. The lines may be coincident. If this is the case, the equations have infinitely numerous solutions [dependent (consistent) pair of equations].

Thus, the graphical representation of simultaneous linear equations in two variables will be in the following forms.

Practice Exam Questions

Solved Examples on Graph of a Linear Equation

Q.1. Represent the linear equation \(2(5y – 2) = 4y + 8\) on the graph.
Ans: The given equation is \(2(5y – 2) = 4y + 8\)
\( \Rightarrow 10y – 4 = 4y + 8\)
\( \Rightarrow 10y – 4y = 8 + 4\)
\( \Rightarrow 6y = 12\)
\( \Rightarrow y = 2\)
Now, the graphical representation of the line \(y = 2\) is given below.

Q.2. Ten students of class \(IX\) had participated in a science quiz. Suppose the number of girls is \(4\) more than the number of boys. Represent the situation graphically.
Ans: Take the number of girls as \(x\) and the number of boys as \(y\).
It is provided that a total of ten students have taken part in the quiz.
\(\therefore \) Number of school girls \( + \) Number of school boys \( = 10\)
i.e. \(x + y = 10 – – – – (i)\)

According to the question number of girls is \(4\) more than the number of boys.
\(\therefore \) Number of girls \( = \) Number of girls \(+4\) 
i.e. \(x = y + 4\) or \(x – y = 4 – – – – (ii)\)
Now, the two solutions of the equation \((i)\) are

and similarly, the two solutions of the equation \((ii)\) are

Now, we plot the points \(A(10,\,0),\,B(0,\,10),\,P(4,\,0)\) and \(Q(0,\, – 4)\) corresponding to these solutions on the graph paper and draw the lines \(AB\) and \(PQ\) representing the equations   \(x + y = 10\) and \(x – y = 4\) as shown below:

Q.3. The cricket coach of a cricket team is buying \(3\) bats and \(6\) balls for \(₹3900\). Later, he is buying another bat and \(3\) more balls of the same kind for \(₹1300\). Represent the situation geometrically.
Ans: Let the price of a bat be \(₹x\) and that of a ball be \(₹y\).
It is given that \(3\) bats and \(6\) balls are brought for \(₹3900\).
\(\therefore \,3x + 6y = 3900 – – – (i)\)
It is also given that one bat and \(3\) balls of the same kind cost \(₹1300\).
\(\therefore \,x + 3y = 1300 – – – (ii)\)
To acquire the equivalent graphical representation, we find two points on the line representing every equation. Specifically, we find two solutions to every equation. we have
\(3x + 6y = 3900\)
When \(y = 0\), we have
\(3x + 0 = 3900 \Rightarrow x = 1300\)
When \(x = 0\), we have
\(0 + 6y = 3900 \Rightarrow y = 650\)
Thus, two solutions of the equation \((i\)) are:

\(x + 3y = 1300\)
When \(y = 100\), we have
\(x + 300 = 1300 \Rightarrow x = 1000\)
When \(x = 100\), we have
\(100 + 3y = 1300 \Rightarrow y = 400\)
Thus, two solutions of the equation \((i\)) are:

Now, we plot the points \(A(1300,\,0)\) and \(B(0,\,650)\) and drawn the line \(AB\) passing through these two pints to represent equation \(3x + 6y = 3900\). To represent the equation \(x + 3y = 1300\), we plot the points \(P(1000,\,100)\) and \(Q(100,\,400)\) and drawn line \(PQ\) passing through these points.

We observe that the two lines representing the two equations are intersecting at the point \(A(1300,\,0)\)

Q.4. Romila went to a stationary stall and purchased \(2\) pencils and \(3\) erasers for \(₹9\). Her friend Sonali to have seen the new variety of pencils and erasers with Romila, and she also bought \(4\) pencils and\(6\) erasers of the same kind for \(₹18\). Represent this situation graphically.
Ans: Let the cost of \(1\) pencil be \(₹x\) and the cost of eraser be \(₹y\).
It is given that Romila purchased \(2\) pencils and \(3\) erasers for \(₹9\).
\(\therefore \,2x + 3y = 9 – – – (i)\)
It is given that Sonali purchased \(4\) pencils and \(6\) erasers for \(₹18\).
\(\therefore \,4x + 6y = 18 – – – (ii)\)
To acquire the equivalent graphical representation, we find two points on the line representing every equation. That is, we are going to find two solutions to every equation. we have
\(2x + 3y = 9\)
Putting \( x=-3 \), we get
\( – 6 + 3y = 9 \Rightarrow 3y = 15 \Rightarrow y = 5\)
Putting \( x=0 \), we get
\(0 + 3y = 9 \Rightarrow 3y = 9 \Rightarrow y = 3\)
Thus, two solutions of \(2x + 3y = 9\) are:

We have,
\(4x + 6y = 18\)
Putting \( x=3 \), we get
\(12 + 6y = 18 \Rightarrow 6y = 6 \Rightarrow y = 1\)
Putting \( x=-6 \), we get
\( – 24 + 6y = 18 \Rightarrow 6y = 42 \Rightarrow y = 7\)
Thus, two solutions of \(4x + 6y = 18\) are

Now, we plot the points \(A( – 3,\,5)\) and \(B(0,\,3)\) and draw the line passing through these points to obtain the graph of line \(2x + 3y = 9\). Points \(P(3,\,1)\) and \(Q( – 6,\,7)\) are plotted on the graph paper, and we join them to obtain the graph of line \(4x + 6y = 18\). We find that the two lines \(AB\) and \(PQ\) coincide.

Q.5. Check graphically to determine if the pair of equations \(x + 3y = 6\) and \(2x – 3y = 12\) is consistent. If so, solve them graphically.
Ans: Let \(x + 3y = 6 – – – (i)\)and \(2x – 3y = 12 – – – (ii)\)
Let us get the graphs of the equations \((i\)) and \((ii\)). For this, we find two solutions to each of the equations. That is,

By plotting the points \(A(0,\,2),\,B(6,\,0),\,P(0,\, – 4)\) and \(Q(3,\, – 2)\) on graph paper, and joining the points to form the lines \(AB\) and \(PQ\), we get

From the graph, a point \(B(6,\,0)\) common to both the lines \(AB\) and \(PQ\). Thus, the solution of the couple of linear equations is \(x = 6\) and \(y = 0\).
Therefore, the given pair of equations is consistent.

Summary

In this article, we learnt about the definition of the graph of a linear equation, the graph of a linear equation notes, graphical representation of pair of linear equations in two variables, solved examples on the graph of a linear equation. The graph of the linear equation is a collection of points in the coordinate plane that all are solutions for the equation. The learning outcome of this article is, the graphical representation of linear graphs helps us to understand the relation between the linear equations.

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FAQs on Graph of a Linear Equation

Q.1. How to solve a linear equation in two variables?
Ans: Solution of a linear equation means a pair of values, one for \(x\) and one for \(y\) which satisfy the given equation.
Example: \(2x + y = 0\)
When we take \(x = 0\), we get \(y = 0\), therefore \((0,\,0)\) is the solution of the equation. When we take \(x = 2\), we get \(y = – 4\), therefore \((2,\, – 4)\) is the solution of the equation. When we take \(x = 3\), we get \(y = – 6\), therefore \((3,\, – 6)\) is the solution of the equation.
Thus, there is absolutely no end to the various solutions of a linear equation in two variables. Specifically, a linear equation in two variables possesses infinitely many solutions.

Q.2. What is the solution of a linear graph?
Ans: The solution of a linear system is the ordered pair that is a solution to all equations in the system. One way of finding a solution to a linear system has been by graphing. The solution to the system will then be at the point in which the two equations intersect.

Q.3. Explain the graph of a linear equation.
Ans: The graph of the linear equation is a collection of points in the coordinate plane that all are solutions for the equation. Unless all variables represent real numbers, one can graph the equation by plotting enough points to recognize a pattern and then connect the points to include all points.

Q.4. How can you tell if a graph is linear or nonlinear?
Ans: Difference between linear and nonlinear functions. If we plot a linear function, you will get a straight line. There are also nonlinear functions. If you graph the coordinates of a nonlinear function, you will not get a straight line.

Q.5. What is the graph of a linear function?
Ans: Linear functions are the ones for which the graph is a straight line. A linear function has the following form \(y = f(x) = a + bx\). A linear function needs one independent variable and one dependent variable. The independent unknown is \(x\) and the dependent unknown is \(y\).