Simultaneous Equations

An equation in mathematics is a mathematical statement stating that two objects should be equal. Two expressions on each side of an equal sign (=) make up an equation. There are two or more variables in it. In other words, the L.H.S value must equal the R.H.S value. It is necessary to verify the equivalence of an equation while substituting the values of the variables. In mathematics, there are various forms of equations. 

Simultaneous equations:

Two or more quantities are related by two or more equations in a simultaneous equation. It consists of a small number of independent equations. Simultaneous equations are sometimes known as systems of equations since they are made up of a finite number of equations for which a common solution is sought. To solve the equations, we must first determine the values of the variables involved. 

A system of equations, often known as a set of simultaneous equations, can be characterised as follows: 

1. Simultaneous linear equations

2. Simultaneous non-linear equations

3. System of bilinear equations

4. Simultaneous polynomial equations

5. System of differential equations

Methods for solving simultaneous equations:

Various approaches can be used to solve simultaneous linear equations. Substitution, elimination, and the augmented matrix method are three alternative approaches of solving simultaneous equations. The two simplest strategies will successfully solve the simultaneous equations to obtain correct solutions among these three ways. We will explain these two fundamental strategies in this article, namely, 

1. Elimination method

2. Substitution method

3. Comparison method

Apart from those methods, Cramer’s rule can be used to solve a system of linear equations. If the simultaneous linear equations have only two variables, we can solve them using the cross-multiplication approach. 

Elimination method:

Example: Using the elimination method, solve the following simultaneous equations. 

4a + 5b = 12

3a – 5b = 9 

Solution: The given equations are, 

4a + 5b = 12…….(1)

3a – 5b = 9……….(2)

Variable ‘b’ has the same sign as the other equation and has the same coefficient. To get rid of the variable ‘b,’ add equations 1 and 2. 

The terms that are similar will be added.

(4a + 3a) + (5b – 5b) = 12 + 9

7a = 21

Add the coefficient of a to the equation’s R.H.S. 

a = 21/7

We get a = 3 by dividing the R.H. S of the equation. Substituting the value a=3 into equation (1), we get 

4(3) + 5b = 12

12 + 5b = 12

5b = 12 – 12

5b = 0

b = 0/5 = 0 

As a result, a = 3 and b = 0 is the solution to the given simultaneous equations. 

Substitution method:

Example: Using the substitution approach, solve the following simultaneous equations. 

b = a + 2

a + b = 4 

Solution: The given equations are as follows,

b = a + 2……..(1)

a + b = 4………(2) 

In the second equation, substitute the value of b. We’ll get it, 

a + (a + 2) = 4

Solving for a, 

a + a + 2 = 4

2a + 2 = 4

2a = 4 – 2 = 2 

a = 2/2 = 1

In equation 1, replace a with this value.

b = a + 2

b = 1 + 2

b = 3 

As a result, the given simultaneous equations have the following solution: a = 1 and b = 3. 

Comparison method:

1. Find an unknown quantity of values in respect to other equations using the first equation. 

2. Find the same unknown quantity of values in connection to the other equations from the second equation. Consider a set of Simultaneous Equations at x and y. The value of x is represented by y in the first expression. The value of x is also represented by y in the second statement. 

3. Since x = x and y = y for all values of x and y, the two values obtained are plainly the same. Equate these two values to create an equation for one variable, then solve it. 

4. Find the value of the second variable by substituting the value of one of the variables into one of the previous equations. 

Conclusion:

Linear equations are frequently used in financial domains. To balance accounts, calculate pricing, and set budgets, accountants, auditors, budget analysts, insurance underwriters, and loan officers employ equations. Athletes and cyclists may calculate the ideal routes for their daily practice routine using the three important variables of speed, distance, and time. They can use different Mathematical formulas to achieve various goals, such as running the longest distance, increasing endurance, or increasing speed.