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. 

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. 

Understanding Simultaneous Equations

Simultaneous Equations are two equations, each with the same two unknowns, and require algebraic skills to find the character values ​​of two or more equations. They are called Simultaneous Equations because the equations are solved together at the same time.

Let us consider the equation x - y = 2 , where both x and y are unknown. There are an infinite number of solutions to this equation, as every pair of numbers with a difference of 2 satisfies the equation. For example, x=5, y=3 ; x=3, y=1; x=4, y=2; x=7, y=5; x=8, y=6; x=10, y=8; and so on. However, if, together with this equation we consider the equation x + y= 8 then x and y must be such that their difference is 2 and their sum is 8. Thus, the two equations x - y= 2, x + y= 8 will both be satisfied by the same pair of values of x and y only when x= 5 and y= 3.  

Methods of Solving Simultaneous Equations 

There are three common methods in which you can solve the Simultaneous Equations in the method that is apt for the question and sometimes it may not be the one you prefer but you still got to do it. 

1. Substitution Method

Step 1: From the two equations given, consider one of the equations from the question and then find the value of one variable in relation to another variable. 

Step 2:  Substitute the value of the variable obtained in step 1 above into another equation and solve it to get the value of one variable. 

Step 3:  Substituting the value ​​of the variable obtained in step 2 above will result in the results of step 1 getting the values ​​of the remaining unknown variable. And this way you find the values of all the unknown variables. 

2. Elimination Method 

Step 1: Multiply one or both  equations by one or more appropriate numbers so that either the x or y coefficients of both equations are numerically the same. 

Step 2: Add or subtract one equation to the other so that  terms with the same numeric coefficients cancel each other out. All that remains is either the x or y only equation. 

Step 3: Solve the resulting equation to find one value of the unknown. 

 Insert this value into one of the two given equations to find the  other unknown value. 

3. Comparison Method 

Step 1: From the first equation, find an unknown quantity  of values ​​in relation to other equations. 

Step 2: From the second equation, find  the same unknown quantity  of values ​​in relation to the other equations. Suppose you have a pair of Simultaneous Equations at x and y. From the first expression, the value of x is represented by y. Also, from the second expression, the value of x is represented by y. 

Step 3: The two values ​​thus obtained are clearly the same, since x = x and y = y for all values ​​of x and y. Equate these two values ​​to get an equation for one variable and solve  that variable. 

Step 4: Substitute the value of one of the variables into one of the previous equations and find  the value of the other variable.

4. 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.