Introduction: Linear Equation in Two Variables | Mathematics for EmSAT Achieve PDF Download

A Linear equation is defined as an equation with the maximum degree of one only, for example, ax = b can be referred to as a linear equation, and when a Linear equation in two variable comes into the picture, it means that the entire equation has 2 variables present in it. Therefore, Linear Equation in two variables can be written in the general form of, ax + by + c = 0, where a, b, c are the constants and x, y are the variables.

Linear Equation in Two Variables

A linear equation in two variables is given by,
Ax + By + C = 0
Where, A, B, and C are constant real numbers, and A and B both are never zero.
Let’s see how to formulate such an equation through a real-life example.

Question: A one-day international match was played between Australia and India in Nagpur. Two Indian batsmen scored a total of 176 runs. Express this information in the form of an equation.
Answer:
We know that two batsmen scored 176 runs, but we do not know how much each batsman scored. Let the runs scored by each batsman be “x” and “y”.
So, x + y = 176.
This is the required linear equation in two variables.
Linear equations in one variable can also be represented as a linear equation in two variables. For example: x = 3, it can also be re-written as
x.1 + y.0 = 3

Solutions of Linear Equation in two Variables

We have seen equations such as x = 5, y = 10. They have only one solution. But when it comes to linear equations in two variables. There are more than one solution,

For example: Let’s assume an equation in two variables,
x + 2y = 4
To find the solutions to this equation, we need to know the values of x and y which satisfy this equation. Here x = 2 and y = 1 is a solution, let’s verify it. Plug in the values of x and y in the equation given above.
x + 2y = 4
⇒ (2) + 2(1) = 4
⇒ 2 + 2 = 4
Hence, x = 2 and y = 1 is a solution this equation. Similarly, we can also verify that x = 2 and y = 1 is also a solution. We can do more solution like this by simply assuming a value of x and then putting in the equation. For example: let’s assume x = 4. Now plug it in the equation, the equation reduces to a single variable equation.
4 + 2y = 4
⇒ 2y = 0
⇒ y = 0
So, if we keep on taking different values of x, we can find infinitely many solutions to these equations.
Thus, we can say, Linear Equations in two variables have infinitely many solutions.

Question 1: Find three different solutions to the equation y + 5x = 10.
Solution:
To find different solutions, we simply have to assume a value of x or y. Plug it in the equation and reduce it in a single variable equation. This way we can find the value of other variable.
Let’s say x = 2. Plug it in the equation,
y + 5(2) = 10
⇒ y + 10 = 10
⇒ y = 0
So, (2,0) is a solution.
Now let’s say x = 3 and plug it in the equation,
y + 5(3) = 10
⇒ y + 15 = 10
⇒ y = 10 – 15
⇒ y = -5
The solution comes out to be (3, -5)
For the last required solution assume x = 0 and plug it in,
y + 5(0) = 10
⇒ y = 10
The solution comes out to be (2, 10)
Thus, the three solutions are :- (2, 0); (3, -5) and (2, 10). 

Graph of Linear Equation in Two Variables
So far we have seen solutions of linear equations. There are infinitely many solutions to a linear equation in two variables. Let’s look at their geometric interpretation. We can show all the solutions in a coordinate plane and see what it looks like. Let’s see how to do this.
Let’s take an example, x + 2y = 6.
Its solutions can be arranged in the form of the table given below.
x    0    2    4    6
y    3    2    1    0
Let’s plot (0, 3); (2, 2); (4, 1) and (6, 0) on the graph.

Notice that all these points when joined form a straight line. Every point on this line satisfies the equation and every solution of this equation is on the line. This is called a graph of the linear equation. To plot the graph of a linear equation we require a minimum of two solutions of the equation.

Graphing Linear Equation by Plotting Points

1. Find three points whose coordinates are solutions to the equation.
2. Plot the points in a graph.
3. Draw a line through all three points.

Let’s take an example,

Question: Plot the equation of line 2x + y = 4 using the method mentioned above.
Solution:
First let’s find three points which satisfy this equation. This is done by the method previously discussed
Let’s put x = 0, then y comes out to be y = 4. So (0,4) is one point.
Putting y = 0, we get, x = 2. So (2,0) is another point.
Putting x = 1, we get y = 2. So (1,2) is the third point.

x    y
0    4
2    0
1    2
Let’s plot the points given in the table and join them to form a line.

The figure represents our required line.

Equation of lines parallel to x-axis and y-axis
Let’s consider the equation x = 3.
Now when this equation is treated as a single variable equation, it has only solution x = 3. But when we treat it as an equation in two variables
x + 0.y = 3. This equation has infinitely many solutions. Let’s plot graphs of such equations. 

Question 1: Plot the equation of x = 4.
Solution:
Let’s find the solution for this equation
x + 0.y = 4
x = 4 satisfies the equation, and we can put any value in place of y, it won’t affect the solution. Thus, the solutions to this equation look like this,
x    4    4    4
y    0    1    2

Let’s plot this points on graph and get the line.

Question 2: Plot the graph of y = 3.
Solution:
Let’s find the solution for this equation
0.x + y = 3
y = 3 satisfies the equation, and we can put any value in place of y, it won’t affect the solution. Thus, the solutions to this equation look like this,
x    0    1    2
y    3    3    3

Let’s plot this points on graph and get the line.

Question 3: A taxi fare in Bangalore is as follows: For the first kilometer it is Rs.10 and then the subsequent distance is measured in Rs.5 per kilometer. Formulate the linear equation for this problem and draw its graph.
Solution: 
Let the total distance traveled by “x” Km and the total fare be “y”.
y = 10 + (x – 1)5
⇒ y = 10 + 5x - 5
⇒ y = 5x + 5
Now let’s find the solutions to this equation and plot it.
x    -1    0    2
y    0    5    15
The table represents three solutions of the equation. Let’s plot these points on the graph and join them to make a straight line.

Question 4: Rahul and Ravi contributed a total of Rs 100 towards the Covid Relief Fund set up by the government. Formulate the equation which satisfies the data and plot its graph.
Answer: Let the contribution of Rahul be Rs. x and that of Ravi be Rs. y.
Then equation can be formed as,
x + y = 100
Again, let’s find some solutions to this equation.
x    20     40     60
y     80    60    40
Plotting the points on the graph.