Simultaneous Equations

Simultaneous equations are two or more mathematical equations that have the same unknown variables, such as x, y, etc. These equations are linked, and we determine the values of variables that satisfy all equations simultaneously. The simultaneous equations have a common solution.

Given below are some examples of simultaneous equations.

While solving simultaneous equations, we have to choose the right method depending on the type of equations we have. The two common types of equations involved in simultaneous systems are:

1. Linear Equations
    
2. Quadratic Equations

• Linear Equations
  
  Variables like x and y that do not have powers greater than one are called linear variables. Equations with such variables are linear equations. When we have two linear equations together, it is called a system of linear simultaneous equations.
  
  Examples of linear equations include x + y = 10, x - y = 2. We can solve these equations using the elimination or substitution method. 

• Quadratic Equations:
  
  The equation in which the power of the variable is raised to two or squared is known as a quadratic equation.
  
  x² + x + 2 is an example of a quadratic equation. When we have one quadratic equation and one linear equation, it is called a quadratic simultaneous equation. 

Finding the exact values of the variables that make both equations true at the same time is simultaneous equations. There are three main methods for solving simultaneous equations, they are:

1. Substitution Method
    
2. Elimination Method
    
3. Graphical Method 

In the substitution method, we choose one equation and make one variable the subject; we will find out the value of the other variable, like x = something. Put that x into another equation to find the value of the second variable, and then find the value of the first variable.

Let's take a practice problem to better understand this method.

Practice Problem: Solve x + y = 4 and 2x - 3y = 9

Solution: We can derive the value of x from the first equation.

\(x + y = 4\\ x = 4 - y\)
 

1. Put the value of x in the second equation,
   
   \(2x - 3y = 9 \\ 2(4 -y) - 3y = 9\\ 8 - 2y - 3y = 9\\ 8 - 5y = 9\\ -5y = 9 - 8\\ -5y = 1\\ y = -⅕\)
    
2. Now put y = -⅕ in the first equation,
   
   \(x + y = 4 \\ x - ⅕ = 4\\ x = 4 + ⅕\\ x = 21/5\)

Therefore, the value of x = 21/5, y = -⅕.

In the elimination method, we make the coefficients the same and then add or subtract the equations to cancel one variable. Then, solve for one variable and substitute to get the value for the other variable.

Let's practice this using an example.

Example: Solve 2x - 5y = 3 and 3x - 2y = 5

Solution: We can make the coefficient of y the same.
 

1. Multiply 2 by the first equation
   \(4x - 10y = 6\)
    
2. Multiply 5 by the second equation
   \(15x - 10y = 25\)
    
3. Now subtract both 
   \((15x - 10y) - (4x - 10y) = 25 - 6\\ 11x = 19\\  x = 19/11\)
    
4. Now, substitute the value of x into any of the original equations.
   \(2x - 5y = 3\\ 2 (19/11) - 5y = 3\\ 38/11 - 5y = 3\\ -5y = 3 - 38/11\\ -5y = 33 - 38 /11\\ -5y = -5/11 \\→ {y = {(-5) \over {11 (-5)}} }\\→ y = 1/11\)

The values are, x = 19/11, y = 1/11

In the graphical method, we draw both equations as lines on a graph, and the point of intersection is the solution. There are three possible outcomes in a graphical method: they are

• One point of intersection - When the system has only one solution.

• Lines overlap completely - When the lines overlap completely on all the points in a system, then the system has infinite solutions. 

• No point of intersection: When the lines are parallel, the system has no solution.

Example: Solve x + y = 10, x - y = 4

Solution:

1. For x + y = 10:
    

   If x = 0, y = 10 

   If x = 10, y = 0
   
   Points are: (0, 10) and (10, 0)  
    

2. For x - y = 4
   
   If x = 0, y = -4
   
   If x = 4, y = 0
   
   Points are: (0, -4) and (4,0)
    

3. When we draw both the lines on the graph, they meet at (7, 3)

The solution is x = 7, y = 3.

For students in smaller grades, learning simultaneous equations can be difficult. To make this easy for children and parents, here are a few essential tips and tricks:

1. Children can use calculators like simultaneous equation calculator to verify their final answers.
    
2. To eliminate a variable, make the coefficients equal before adding or subtracting equations.
    
3. If the equations have coefficients in decimals, multiply it by powers of 10 for easy calculation.
    
4. Remember, when using a substitution method, substitute the value of the variable obtained from one equation into the another equation.
    
5. Write the calculated value of x and y in the table to avoid mistake when plotting the graph.

Parent Tip: Encourage your child to solve problems daily. Relate problems with real life objects like food, toys, snacks to make it fun. Use can also use online calculators to verify the answers.

Simultaneous equations are not only used in math problems and equations, but also it is used in many real-life applications. Here are a few real-life applications.

1. Business and Finance
   
   In business, simultaneous equations are used to calculate the costs, profits, or pricing of the products. For example, a shop sells two types of T-shirts. The total sales from both types is $500, and 40 T-shirts were sold in total. Simultaneous equations help in figuring out how many of each type were sold.
   
    
2. Engineering
   
   In engineering, it is used to solve unknowns in electrical circuits, forces, or mechanical systems. Ohm’s law and Kirchhoff’s law in electrical engineering give equations for current and voltage in circuits. Simultaneous equations are used to find the current through each part of a complex circuit
   
    
3. Computer Science and Coding
   
   Simultaneous equations are used for data analysis, and graphics like 3D modelling. It helps to determine where two objects in a game or animation will collide or intersect.
   
    
4. Animations and Graphics
   
   To model motion, throwing or movements of objects in 2D planes, equations are used to program algorithms. Simultaneous equations are, hence, widely used in animation, video games and computer graphics.
   
    
5. Mixing Proportions
   
   Simultaneous equations are used in cooking, industries, chemistry to mix different items or ingredients in a desired proportion.