Simultaneous Linear Equations in Two Variables

Linear Equations

An equation is said to be linear if the degree of the variable is 1. For example, x + 7 = 0, 3x + 4y = 12, x + y = 5 are linear equations.

Linear Equation in Two Variables

An equation of the form ax + by = c, where a, b, c, are constants and a, b ≠ 0 is a linear equation in two variables x and y.

Simultaneous Linear Equations

Two linear equations in two variables which are satisfied by the same pair of values of the variables are called simultaneous linear equations in two variables.

For example, 2x – y = 7 and 3x – 2y = 9.

The values of x and y which satisfy both the equations are called the solutions of the simultaneous linear equations. Here, in the above example if we put x = 5 and y = 3, we observe that both the equations are satisfied. Thus, we say that x = 5 and y = 3 is the solution set of the given equations.

Note that the solution set of the pair of simultaneous linear equations is unique.

Methods of Solving Simultaneous Linear Equations

There are three methods for solving a pair of simultaneous linear equations in two variables:

1. Substitution Method

2. Elimination (or Addition-Subtraction) Method

3. Comparison Method

Substitution Method

In order to solve a pair of simultaneous linear equations in two variables, proceed as follows:

1. Find the value of one of the variable by solving any one of the given equations.

2. Substitute that value of the variable in the other equation.

3. Now, solve the required linear equation in one variable and substitute this value into either of the two original equations and solve it to find the value of the other variable.

Example: Solve the following pair of simultaneous linear equations:

3x + 5y = 28

x – 7y = –34

Solution: Given equations are

3x + 5y = 28 … (1)

x – 7y = –34 … (2)

Let us first find the value of one of the variable from any one of the given equations. In order to avoid fractions, let us express x in terms of y using the equation (2)

i.e., x – 7y = –34 ⇒ x = –34 + 7y

Now, putting x = –34 + 7y in equation (1), we get 3 × (−34 + 7y) + 5y = 28

⇒ –102 + 21y + 5y = 28 ⇒ 26y = 130

⇒ y = 130/26 ⇒ y = 5

Now, putting this value of y in equation (2), we have x – 7 × 5 = −34 ⇒ x = −34 + 35 = 1

∴ the solution set is x = 1 and y = 5.

Elimination or Addition-Subtraction Method

In order to avoid substituting fractions in solving equations, we use elimination method. To solve the pair of simultaneous linear equations, proceed as follows:

1. Transform the equations by multiplying one or both the equations to eliminate one variable by adding or subtracting.

2. Now, solve the required linear equation in one variable and substitute this value into either of the two original equations and solve it to find the value of the other variable.

Example: Solve: 3x + 4y = 11 ... (1)

x + 7y = 15 ... (2)

Solution: If we multiply equation (2) by 3, we shall have the same coefficient of x in both equations.

The new equation is 3x + 21y = 45 ... (3)

We can now eliminate x by taking away, i.e., subtracting (1) from equation (3)

3x + 21y = 45 ... (3)

3x + 4y = 11 ... (1)

17y = 34 Þ y = 2

We substitute y = 2 in equation (2), we have x + 14 = 15 Þ x = 1.

Hence, the solutions are x = 1 and y = 2.

To check these values, substitute them in equation (1), we have

L.H.S. = 3 ´ 1 + 4 ´ 2 3 + 8 = 11 = R.H.S.

As the L.H.S. = R.H.S. the solutions are correct.

Comparison Method

In order to solve a pair of simultaneous linear equations in two variables, proceed as follows:

1. Transform both the equations in terms of one of the same variable.

2. Equate the values so obtained and then solve the required linear equation.

3. Substitute the value of the variable in either of the two equations and solve it to fi nd the value of the other variable.

Example: Solve the following pair of equations.

2x + 3y = 2 … (1)

4x + 5y = 6 … (2)

Solution: Given equations are:

2x + 3y = 2 … (1)

4x + 5y = 6 … (2)

Expressing equation (1) and (2) in terms of one variable say x, we get

x = (2 – 3y)/2 … (3) and x = (6 – 5y)/4 … (4)

Now, equating equation (3) and (4) and then solving, we get

(2 – 3y)/2 = (6 – 5y)/4

⇒ 2(2 – 3y) = 6 – 5y ⇒ 4 – 6y = 6 – 5y

⇒ –6y + 5y = 6 – 4 ⇒ –y = 2 ⇒ y = –2

Putting y = –2 in equation (3), we get x = 4

∴ the solution is x = 4 and y = –2.

Two more methods are also used to solve the simultaneous linear equations in one variable.

1. Graphical method

2. Cross-multiplication method

Graphical Method of Solving Simultaneous Linear Equations

Example: Solve by the graphical method: x + 3y = 6, 2x – 3y = 12.

Solution: From first equation, y = (6 – x)/3

x	0	6
y = (6 – x)/3	2	0

So, the points are (0, 2) and (6, 0).

From second equation, y = (2x – 12)/3

x	0	3
y = (2x – 12)/3	-4	-2

So, the points are (0, -4) and (3, -2).

Now, plot these points for both the equations on the same graph paper.

From the graph, we find the common point of intersection, i.e., (6, 0).

So, the solution of the equations is (6, 0).

Cross-multiplication Method of Solving Simultaneous Linear Equations

The general form of a pair of linear equations in two variables is:

a₁x₁ + b₁y₁ + c₁= 0 … (1)

a₂x₂ + b₂y₂ + c₂= 0 … (2)

Write the coefficients of x₁, y₁, x₂ and y₂ as follows:

b₁ c₁ a₁ b₁

b₂ c₂ a₂ b₂

Now, cross multiply and write as follows:

x/(b₁c₂ – b₂c₁) = y/(c₁a₂ – c₂a₁) = 1/(a₁b₂ – a₂b₁)

Now, x = (b₁c₂ – b₂c₁)/ (a₁b₂ – a₂b₁) and y = (c₁a₂ – c₂a₁)/ (a₁b₂ – a₂b₁)

Example: Solve for x and y by cross-multiplication method:

2x – y = 6 and x – y = 2

Solution: 2x – y = 6 . . . (1)

x – y = 2 . . . (2)

By cross-multiplication method we have

Nature of Simultaneous Linear Equations