Simultaneous Linear Equations

A linear equation in algebra is an equation having the highest degree of 1. It represents straight-line relationships and does not include powers, roots, or products of variables. For example, x + y = 10 is a linear equation. In its standard form, it is written as ax + by = c.
 

Simultaneous linear equations are a set of two or more linear equations that have the same set of variables, like x and y. They are used to find the values of these variables that satisfy all equations simultaneously.
The general form of a system with two variables is,
a1x + b1y = c1
a2x + b2y = c2
Where x and y are variables and a, b, and c are constants.
 

There are three methods of solving simultaneous linear equations:

Substitution method

The substitution method is most suitable for simple equations or equations that can easily be rearranged. Any one of the equations is used to solve for a variable. That value is substituted into the other equation to find the value of the other variable. 
For example, let's take two equations, (i) and (ii)
x + y = 10  (i)
x - y = 4     (ii)
We can find the value of x in terms of y using equation (i) 
x + y = 10
x = 10 - y
Now, we can substitute this value of x in equation (ii) to find y
x - y = 4
(10 - y) - y = 4
10 - 2y = 4
2y = 6
y = 3
So, x = 10 - y = 10 - 3 = 7
Answer: x = 7, y = 3
 

Elimination method

This method requires addition or subtraction operations to cancel out one variable and solve for the other. It is useful for equations in standard form, i.e., ax + by = c, and quicker than the substitution method for equations that are not easily rearranged.
For example, for a set of two equations,
2x + 3y = 12  (i)
2x - y = 4    (ii)
We subtract (ii) from (i)
(2x + 3y) - (2x - y) = 12 - 4
2x - 2x + 3y + y = 8
Here, x is eliminated, so we get the value of y
4y = 8
y = 2
Now, we can use this value of y to find x using any equation
Substitute y = 2 in equations 2x - y = 4
2x - 2 = 4
2x = 6
x = 3
Answer: x = 3, y = 2

Graphical method 

The graphical method helps visualize the position of the point of intersection between the two lines. The points where the two lines intersect on a graph are the solutions of the equations. If the lines intersect at one point, the system has one unique solution and is consistent and independent. If the lines are parallel, the system has no solution and is inconsistent. If the two lines completely overlap, then the system has infinitely many solutions, and is dependent.
For example,
x + y = 6   (i)
x - y = 2    (ii)
Convert both equations to the form y = mx + c
y = 6 - x
y = x - 2
Make a table of values of both variables x and y, then plot them on the graph. For these values of x, we find the value of y by substituting them in the equations.

Both the lines intersect at (4,2), so, x = 4 and y = 2.
 

Tip 1: Before choosing a method to solve the equation, always simplify both equations.
Tip 2: Substitution is useful when one of the variables is easily isolated.
Tip 3: Elimination is better when the coefficients are easily matched
Tip 4: For the standard form of equations, use the cross multiplication method.
Tip 5: Labelling each step helps avoid confusion during elimination and substitution.
 

Simultaneous linear equations are practical tools used in many real-life situations to analyze and solve multivariable problems efficiently.

• Cost and profit analysis in business and economics
  Simultaneous equations help determine the break-even point, which means that the cost = revenue. For example, if x units of a product cost C = 50x + 1000 and the revenue is R = 70x, then C = R gives us the value of x, which determines whether the business is at a profit or loss. 
• Here, 50x + 1000 = 70x 
• 1000 = 20x
• x = 50, so the break-even point is 50 units.
  
   
• Circuit analysis in engineering
  In electrical engineering, Kirchhoff’s laws form simultaneous equations to find current and voltage in different branches of a circuit.
  
   
• Balancing chemical equations
  Balancing complex chemical reactions involves setting up equations where the number of atoms of each element must be the same on both sides.
  
   
• Structural load distribution in construction
  In civil engineering, simultaneous equations are used to calculate load distribution on different beams to ensure structural stability.
  
   
• Route and schedule planning for transportation
  These equations are used by transportation companies to optimize travel routes according to delivery timings and fuel usage.