Any equation of the first degree having two variables such as x and y is called a linear equation with two variables.
Let's write down a linear equation with two variables.
Suppose the cost of two books is $10.
If the cost of the first book is x and the cost of the second book is y, then the equation is x + y = 10, where x and y are variables.
Let's find the solution set for linear equations
Example 1
x + y = 10
If the cost of one book is $1, then cost of other book will be $9 or,
If the cost of the first book is $2, then the cost of the second book will be $8 and so on.
The values that satisfy the equation, i.e., make the equation true, are called solutions or roots of the equation.
The values that make the equation x + y = 10 true are
x = 1 y = 9
x = 2 y = 8
x = 3 y = 7
x = 4 y = 6
x = 5 y = 5 etc.
Therefore, the solution set is
{(1, 9) (2, 8) (3, 7) (4, 6) (5, 5) (6, 4) (7, 3) (8, 2) (9, 1)}
Example 2
In this example, let's assume that x and y are integers.
As you know, a set of all integers is represented by
Z = {… … … -5, -4, -3, -2, -1, 0, +1, +2, +3, +4 … … …}
Therefore, in the equation, x + y = 0 and x, y are integers.
The solution set = {(1, -1) (2, -2) (3, -3) ad infinitum}
The solution for this equation is infinite; therefore, the solution set is an infinite set.
Example 3
If x = 2 and y = 4, determine whether this satisfies the equation
x + y = 5.
By substituting the values of x and y in the equation
x + y = 5
2 + 4 = 6, which is not equal to 5.
The values x = 2, y = 4 do not satisfy the equation; therefore, the set (2, 4) is not a solution set.
Try these questions
Verity whether the values of x and y given against each of the following satisfy the given equation.
- 2x - y = 3 and x = 4, y = 1
Answer: Substituting x, y values in the given equation.
2 (4) - 1 = 8 - 1 = 7
Hence (4, 1) is not a solution set.
- x + y = 9 and x = 3, y = 6
Answer: 3 + 6 = 9 = 9
Hence (3, 6) is a solution set.
- 2x + y = 9 and x = 2, y = 2
Answer: 2 (2) + 2 = 4 + 2 = 6 which is not equal to 9.
Hence (2,2) is not a solution set.
- x + y = 10 and x = 6, y = 7
Answer: 6 + 7 = 13 which is not equal to 10.
Hence (6,7) is not a solution set.
- x + 3y = 6 and x = 1, y = 0
Answer: 1 + 3 (0) = 1 which is not equal to 6.
Hence (1,0) is not a solution set.
- x + y = 7 (x, y) = (4,3)
Answer: 4 + 3 = 7
Hence (4,3) is a solution set.
- 2x + 3y = 5 (x, y) = (1,1)
Answer: 2 (1) + 3 (1) = 2 + 3 = 5
Hence (1,1) is a solution set.
- 2x - y = 7 (x, y) = (1,2)
Answer: 2(1) - 2 = 0 which is not equal to 7.
Hence (1,2) is not a solution set.