Simultaneous Equations

Simultaneous equation is an essential chapter in the Algebra branch of Mathematics. There are times when you come across two or more unknown quantities and two or more equations relating to them. These are called simultaneous equations. When you are asked to solve such equations, you must find values of the unknowns which satisfy all the given equations at the same time. They are simultaneous equations because the equations are solved at the same time. Simultaneous equations need algebraic skills to find the values of letters within two or more equations. 

It is an essential point to remember that these equations include a set of few independent equations. It is the reason why simultaneous equations are also known as the system of equations, in which it consists of a finite set of equations. To work on these means looking out for the standard solution. To solve the equations, we need to find the values of the variables that are part of these equations. 


Simultaneous Equation Method

We can solve equations that have more than one unknown variable with an infinite number of solutions. For example, 2x + y = 10 could be solved by: 

X = 1 and y = 8

X =2 and y = 6

X =3 and y = 4

To be able to solve an equation like this, another equation needs to be in use alongside it. So, in this way, it is possible to find the only pair of values that solve both equations at the same time. Hence, these are known as simultaneous equations.

An example of this is:

3x + y = 11, and

2x + y = 8

The unknowns of x and y have the same value in both equations. This fact can be of use to help solve the two simultaneous equations at the same time and find the values of x and y.


How to Solve Simultaneous Equations 

We can solve Simultaneous equations using more than one method. In mathematics, we have the benefit of three different techniques to solve the simultaneous equations such as substitution, elimination, and augmented matrix methods. Among these three methods, we will discuss the two simplest ways that will solve the simultaneous equations to get precise solutions. Here we are going to discuss the two principal methods i.e. Elimination method and Substitution method 

Let us start solving simultaneous equation with the help of an example. Let us begin with the elimination method first. 


Elimination Method 

Solve the following simultaneous equation 

3x + 4y = 32 ….1 

5x - 4y = 64…..2 

Solution: 

Step1 - To begin, we take both equations and add it to result in 

3x + 4y + 5x - 4y = 64+32

Now we can eliminate one variable y and be left with just one variable, i.e. x

8x = 96 x = 96 / 8 = 12

Step2 – Now that we know x, we can substitute in any of the two equations. So we take the 1st one

3x + 4y = 32 i.e., 3 (12) + 4y = 32 i.e., 36 + 4y = 32

4y = 32-36 i.e., y = -1

So, x = 12 and y=-1


This equation was solved using the elimination method. The elimination or reducing method for solving a pair of simultaneous linear equations shifts one equation to one that has only a single variable. This approach is known as the Gaussian elimination method. 

It is a process which involves removing or eliminating one of the unknowns to leave a single equation which includes the other unknown. 


Substitution Method 

Let us solve the same equation with the substitution approach. 

Now you know that x=8 is part of the solution. So, taking equation (1) or if you wish, you can take (2) in the substitution method. Let us start by substituting this value for x, and this technique will help us to find the value of y. 

3 (8) + 2y = 36 

24 + 2y = 36 

2y = 36 - 24 

2y = 12 

y = 6 

Hence, the full solution to the simultaneous equation is x = 8, y = 6. We can add further that the solution of the pair of simultaneous equations 3x + 2y = 36, and 5x + 4y = 64 is x = 8 and y = 6. It is easily verified by substituting these values into the left–hand side to obtain the values on the right. So, x=8, y = 6 holds true for the simultaneous equations. 


Types of Simultaneous Equations 

When we study Maths, we understand that an equation is a statement in which two things are equal. As you know, equations are algebraic expressions. There are two sides to an expression or equation. Both sides have two or more variables. The LHS and the RHS have to be (=) equal. When values are substituted, there has to be equality. The different types of mathematical equations are- linear, quadratic, and polynomial. 


Real-life Applications of Simultaneous Equations

Financial fields often require the use of linear equations. Accountants, auditors, budget analysts, Insurance underwriters, and loan officers use equations to balance accounts, determine the pricing, and to set budgets. Athletes and cyclists can use the three key variables of speed, distance, and time to calculate the best routes for their daily practice regimen. They can set different mathematical expressions to align with different goals, like running the farthest distance, building endurance, or maximize speed. 

FAQ (Frequently Asked Questions)

1. What is the point of simultaneous equations?  

A simultaneous equation is one where you have two equations with corresponding variables, x or y. It allows you to solve an equation for both the unknowns. Now, this is not possible in other standard single equations, linear or quadratic.  When you work on simultaneous equations, you have to find a solution that holds true for both the equations. This technique can be in use to solve everyday problems, particularly those types of problems that are difficult.  Simultaneous equations is a system that is applicable when it works for two problems or two unknowns.

2. What are simultaneous equations used for?   

Simultaneous equations are in use when deciding the relationship between the price of a commodity and the quantities of the products people want to buy at a certain amount. An equation can be written as one that describes the relationship between supply, price, and other variables such as income.  When you are travelling by plane, car, or a train, you may want to know the values of unknown variables while travelling. These are some cases where simultaneous equations come in handy.