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What are Simultaneous Equations?


Simultaneous equations, also known as a system of equations, are a set of equations that involve multiple variables and are related to each other. The solution to a set of simultaneous equations is the common solution that satisfies all the given equations.

  • Simultaneous equations can be classified into different types based on the type of equations involved. The most common type of simultaneous equations is linear equations, which involve variables raised to the first power and can be represented on a graph as a straight line. Non-linear equations, on the other hand, involve variables raised to a power greater than one and cannot be represented as a straight line.
  • To solve simultaneous equations, we need to find the values of the variables that satisfy all the given equations. There are different methods to solve simultaneous equations, such as substitution method, elimination method, and graphical method. These methods involve manipulating the equations to eliminate one or more variables and then solving for the remaining variables.

The general form of simultaneous linear equations is given as:

ax +by = c
dx + ey = f

Methods for Solving Simultaneous Equations


The simultaneous linear equations can be solved using various methods. There are three different approaches to solve the simultaneous equations such as substitution, elimination, and augmented matrix method. Among these three methods, the two simplest methods will effectively solve the simultaneous equations to get accurate solutions. Here we are going to discuss these two important methods, namely,

1. Elimination Method: In the elimination method, we eliminate one of the variables from the equations by adding or subtracting the equations from each other. The aim is to get one of the variables to have the same coefficient in both equations so that we can add or subtract them easily.
Steps to solve simultaneous equations using the elimination method:

  • Identify which variable to eliminate and write the equations in standard form.
  • Multiply one or both of the equations by suitable constants to get the same coefficient for one of the variables.
  • Add or subtract the equations to eliminate one of the variables.
  • Solve the remaining equation to find the value of one of the variables.
  • Substitute the value of the variable into either of the original equations and solve for the other variable.
  • Check the solution by substituting the values back into both equations.

2. Substitution Method: In the substitution method, we solve one of the equations for one of the variables in terms of the other variable and then substitute that expression into the other equation.
Steps to solve simultaneous equations using the substitution method:

  • Solve one of the equations for one of the variables in terms of the other variable.
  • Substitute the expression obtained in step 1 into the other equation to get an equation in one variable.
  • Solve the equation for the variable.
  • Substitute the value of the variable into either of the original equations and solve for the other variable.
  • Check the solution by substituting the values back into both equations.

Simultaneous Equation Example

Solving Simultaneous Linear Equations Using Elimination Method: Go through the solved example given below to understand the method of solving simultaneous equations by the elimination method along with steps.

Example: Solve the following simultaneous equations using the elimination method.
4a + 5b = 12,
3a – 5b = 9

To solve the simultaneous equations using the elimination method, we need to eliminate one of the variables by adding or subtracting the equations. Here, we can eliminate b by adding the equations together because the coefficients of b are opposites:
4a + 5b = 12
3a – 5b = 9
7a = 21
Dividing both sides by 7, we get:
a = 3
Now, we can substitute the value of a into one of the equations and solve for b. Let's use the first equation:
4a + 5b = 12
4(3) + 5b = 12
12 + 5b = 12
5b = 0
b = 0
Therefore, the solution to the simultaneous equations is:
a = 3, b = 0.

Solving Simultaneous Linear Equations Using Substitution Method: Below is the solved example with steps to understand the solution of simultaneous linear equations using the substitution method in a better way.
Example: Solve the following simultaneous equations using the substitution method.
b= a + 2
a + b = 4.

Using the substitution method, we can substitute the first equation into the second equation:
b = a + 2
a + (a + 2) = 4
Simplifying the second equation:
2a + 2 = 4
2a = 2
a = 1
Now substituting the value of a in the first equation:
b = 1 + 2
b = 3
Therefore, the solution to the given simultaneous equations is a = 1 and b = 3.

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