Overview: Any two linear equations that have only one common sets of solutions is known as simultaneous equation.

Simultaneous Equations

When two linear equations having two variables are taken and their individual solutions are noted down in a table. We find one common pair of solutions in each of the equations. That pair of the common solution is the solution of the given two equations. Thus, any two linear equations having two variables that have only one common sets of solutions is known as simultaneous equation.


Let us understand:

Take a linear equation y = x + 2 and note down the values of x and y when -3 < x < 3 in the table below:
x-2-1012
y01234

Also,
Take another linear equation y = 2x and note down the values of x and y when -3 < x and < 3 in the table below:
x-2-1012
y-4-2024

When we analyse the solutions of above two equations, we come to a conclusion that both of these equations have limitless number of solutions. We could sit down and write its solutions for years. But, comparing the solutions of given two equations, we can only find one single pair of common solution. Here the common solution is (x,y) = (2,4).

What is a simultaneous equation?

A simultaneous equation is a pair of two linear equations that have a common pair of solution which satisfies both the given equations.


Remember in Simultaneous Equations

Here are the important things you need to remember in simultaneous equations:
  1. The given two equations have two variables generally x and y.
  2. The equations are always of the first degree i.e. they have the power to the 1.


Solving a pair of Simultaneous Equations:

Remember, we have two linear equations with two variables in simultaneous equations. We can solve such equations via two different methods:
  1. Substitution Method
  2. Elimination Method


Substitution Method:

In this method, we find the value of one variable (x) in terms of another variable (y) using the first equation. Then, we put that value of the variable (x) in the second equation and find the value of the other variable (y). Again, we out value of the known variable (y) in first equation and get the value of the first variable (x).

If the above explanation was a bit difficult to understand, here's a detailed explanation:

Question: Solve each pair of the following simultaneous equations: x - y = 30 and x + y = 10.

Solution:
Given,
x - y = 30 is the first equation
x + y = 10 is the second equation

Step 1: We take the first equation and find the value of x in terms of y.
or, x - y = 30
or, x = 30 + y
So, we get the value of x in terms of y i.e. x = 30 + y.

Step 2: We take the second equation and put the value of x from the first equation here.
or, x + y = 10
or, (30 + y) + y = 10
or, 30 + y + y = 10
or, 30 + 2y = 10
or, 2y = 10 - 30
or, 2y = - 20
So, y = -10
We get the value of y i.e. y = -10.

Step 3: Put value of y from step 2 in first equation to get the value of x.
or, x = 30 + y
or, x = 30 - 10
So, x = 20
We get the value of x i.e. x = 20.

So, we get the required values of x and y from the given pair of two equations i.e. (x,y) = (20,-10).


Elimination Method:

In this method, we make the value of either of the two variables equal in both of the equations. If in the first equation, we have x = 2y +3  and in the second equation we have 2x = y + 3. We do not have equal values of variables in both of the equations. So, we can either multiply the first equation and get 2x to match it with the variable 2x in the second equation and eliminate the variable via subtraction or do vice-versa. In the elimination method, we aim to make the values of one of the variables (eg: x) in both the equations equal. Then subtract the second equation from the first equation. Finally, solve the new equation to get the value of one of the remaining variable (since x is eliminated, y remains) and put its value in either of the two equations to get the value of the other variable (x). Let us solve the same question from above:

Question: Solve each pair of the following simultaneous equations: x = 2y +3 and 2x = y + 3.

Solution:
Given,
x = 2y +3 is the first equation
2x = y + 3 is the second equation

Step 1: Multiply the first equation so that the value of variable 'x' becomes equal in both the given equations.
or, 2x = 2(2y +3)
or, 2x = 4y +6
Soz the value of variable x becomes equal I'm both the equations.

Step 2: Subtract the second equation from the first equation to get the value of remaining variable 'y'.
or, (2x) - (2x) = (y +3) - (4y +6)
or, 2x - 2x = y +3 -4y -6
or, 0 = -3y - 3
or, 3y = -3
So, y = -1
We get the value of y i.e. y = -1.

Step 3: Put value of the known variable (y = -1) in either of the equations to get the value of the other variable (x). Here, we put the value in first equation,
or, x = 2 (-1) + 3
or, x = -2 + 3
So, x = 1
We get the value of x i.e. x = 1

So, we get the required values of x and y from the given pair of two equations i.e. (x,y) = (1,-1).


Solutions:

Learn more: Here is the link to the solved exercise of Simultaneous Equations from vedanta Excel in Mathematics Book of Class 10.

Subject-wise Notes for Class 10:
Class 10 Mathematics
Class 10 Optional Mathematics
Class 10 Science