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Simultaneous Equations

Before starting with simultaneous equations, let’s recall what are equations in maths and the types of equations. In Mathematics, an equation is a mathematical statement in which two things should be equal to each other. An equation consists of two expressions on each side of an equal sign (=). It consists of two or more variables. In short, the L.H.S value should be equal to the R.H.S value. While substituting the values of the variables in an equation, it should prove its equality. There are different types of equations in Maths, such as:

and so on. In this article, we are going to discuss the simultaneous equations which involve two variables along with different methods to solve.

What are Simultaneous Equations?

The simultaneous equation is an equation that involves two or more quantities that are related using two or more equations. It includes a set of few independent equations. The simultaneous equations are also known as the system of equations, in which it consists of a finite set of equations for which the common solution is sought. To solve the equations, we need to find the values of the variables included in these equations.

The system of equations or simultaneous equations can be classified as:

  • Simultaneous linear equations (Or) System of linear equations
  • Simultaneous non-linear equations
  • System of bilinear equations
  • Simultaneous polynomial equations
  • System of differential equations

Here, you will learn the methods of solving simultaneous linear equations along with examples.

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,

Apart from those methods, we can also the system of linear equations using Cramer’s rule.

If the simultaneous linear equations contain only two variables, we may also use the cross-multiplication method to find their solution.

Simultaneous Equation Example

Let us now understand how to solve simultaneous equations through the above-mentioned methods. We will get the value of a and b to find the solution for the same. x and y are the two variables in these equations. Go through the following problems which use substitution and elimination methods to solve the simultaneous equations.

Try Out: Simultaneous Equation Solver

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

Solution:

The two given equations are

4a + 5b = 12 …….(1)

3a – 5b = 9……….(2)

Step 1: The coefficient of variable ’b’ is equal and has the opposite sign to the other equation. Add equations 1 and 2 to eliminate the variable ‘b’.

Step 2: The like terms will be added.

(4a+3a) +(5b – 5b) = 12 + 9

7a = 21

Step 3: Bring the coefficient of a to the R.H.S of the equation

a = 21/ 7

Step 4: Dividing the R.H. S of the equation, we get a = 3

Step 5: Now, substitute the value a=3 in the equation (1), it becomes

4(3) + 5b = 12,

12 + 5b = 12

5b = 12-12

5b =0

b = 0/5 = 0

Step 6: Hence, the solution for the given simultaneous equations is a = 3 and 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.

Solution:

The two given equations are

b = a + 2 ————–(1)

a + b = 4 ————–(2)

We will solve it step-wise:

Step 1: Substitute the value of b into the second equation. We will get,

a + (a + 2) = 4

Step 2: Solve for a

a +a + 2 = 4

2a + 2 = 4

2a = 4 – 2

a = 2/2 = 1

Step 3: Substitute this value of a in equation 1

b = a + 2

b = 1 + 2

b = 3

step 4: Hence, the solution for the given simultaneous equations is: a = 1 and b = 3

Practice Problems

  1. Solve: 5x + 3y = 7 and -3x + 5y = 23
  2. Solve the following simultaneous equations:
    px + qy – r² = 0

    p²x + q²y – r² = 0

  3. Solve for a and b:
    10a – 8b = 6
    10a – 9b = -2

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