Pair of Linear Equations in Two Variables

Solution of Linear Equations in Two Variables

Now that we know what Linear Equations are and the ways of converting a statement in the form of the linear equation and the various terminologies associated with it. We can discuss the methods of solving linear equations for finding the required solution. Solving linear equations is very simple. Let us get to know the methods to do it!

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Methods for Solving Linear Equations

Linear equations can be solved graphically as well as algebraically. Let us learn about both of them.

Graphical Method

In Graphical method, we draw the lines for the given pair of equations with possible satisfying values on a graph and find out the case being satisfied i.e., whether the drawn lines are intersecting at a point (consistent solution) or are parallel with each other (inconsistent solution) or are coincident (dependent solution).

Algebraic Methods

  • Substitution Method: We substitute one of the given equations in another one by substituting one variable in the form of the other. Now, the equation will contain only one variable and then solve it accordingly to get the desired result.
  • Elimination Method: As the name suggests, in the elimination method, we try to eliminate one of the variables from the given set of equations. Solving it will give us the desired result.
  • Cross-Multiplication Method: The general form of a pair of linear equations in two variables is:

a1x1 + b1y1 = c1 … (i)

a2x2 + b2y2 = c2 … (ii)

In this method, we multiply equation (i) by the coefficient of y(or, x2) i.e., b(or, a2) & equation (ii) by that of  y(or, x1) i.e., b1 (or, a1) & eliminate one of the variables and solve accordingly. The name is given as the multiplication in the equations with the coefficients of the variables are done in a cross fashion.

Check out our detailed article on Linear Equations with 2 variables here.

Solved Linear Equations Examples

Solve by the Graphical method: x + y = 16, & x – y = 4

Solution: The two solutions of each equation

x 2 4 6 8
y = 16 – x 14 12 10 8


x 2 4 6 8
y = x – 4 -2 0 2 4

From the graph, we find the common point of intersection, i.e., (10,6).solving linear equations

More Solved Examples for You

Question: A number consists of two digits such that the digit in the ten’s place is less by 2 than the digit in unit’s place. Three times the number added to 5/7 times the number obtained by reversing the digits equal to 108. What is the sum of the digit of the number?

Solution: Let the two digits of the number be x & y in which x is in the ten’s place & y is in the unit’s place. The number can be written as 10 × x + y  (as 36 = 3 × 10 + 6).

Case 1:  x = y – 2 …(i)
Case 2: The number obtained by reversing the digits = 10 × y + x  (as 63 = 6 × 10 + 3).

A.T.Q., 3(10x + y) + 5 (10y + x) / 7 = 108
⇒ 30x + 3y + 50y / 7 + 5x / 7 = 108
⇒ 210x + 21y + 5x + 50y = 756
⇒ 215x + 71y = 756 …(ii)

Solving (i) and (iii), we get x ≈ 2 & y ≈ 4. The sum of the numbers = 2 + 4 = 6.

Question: The sum of two numbers is 20. Five times one number is equal to 4 times the other. Find the bigger of the two numbers.

Solution: Suppose the two numbers be x & y. We have,

 x + y = 20 …(i)
5x = 4y …(ii)

Multiplying (i) by 5 & subtracting (ii) from it, we get x ≈ 9 & y ≈ 11. The bigger number = 11.

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