Quick Review: Linear Equations

A Linear Equation in one variable is defined as ax + b = 0 or ax = c, where a, b and c are real numbers. Also, a ≠ 0 and x is an unknown variable.
The solution of the equation ax + b = 0 is x = - b/a. We can also say that - b/a is the root of the linear equation ax + b = 0.
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Linear Equation in Two Variables
A Linear Equation in two variables is of the ax + by + c = 0 or ax + by= d type where a, b, c and d are constants and also, both a and b are not equal to 0.
Methods of Solving Two Linear Equations:
  1. Substitution Method
    Step 1: Find the value of one variable say y in terms of the other i.e. x. from either equation.
    Step 2: Then substitute the value of y so obtained in the other equation. Therefore, we have a single equation in one variable x.
    Step 3: Now solve this equation for x.
    Step 4: In the end, substitute the value of x, thus obtained, in first step and find the value of y.
  2. Method of Elimination
    Step 1: Multiply both the equations with such numbers so as to make the coefficients of one of the two unknowns numerically same.
    Step 2: To get an equation containing only one know n, subtract or add the two equations. Solve this equation to get the value of the unknown.
    Step 3: In either of the two original equations, substitute the value of the unknown. Thus, by solving that, the value of the other unknown is obtained.
  3. Short – Cut Method
    Let us consider two equations as: a1x + b1y = c1 and a2x + b2y = c2 Then, the solution will be written as x/(b1 c2 b2 c2 ) = y/(c1 a2 c2 a1 ) = (-1)/(a1 a2 a2 b1 )
    i.e. x = - (b1 c2- b2 c1)/(b1 c2 b2 c1 ) and y = (c1 a2- c2 a1)/(a1 b2- a2 b1 )
Key Points
Suppose, we have two linear equations: a1x + b1y = c1 and a2x + b2y = c2
Then
  • If a1/a2 = b1/b2 , the system will have only one solution that will be consistent. The graphs of this type equation will have intersecting lines.
  • If a1/a2 = b1/b2 = c1/c2 , the system will be consistent with numerous solutions. The graphs of this type of equation will have coincident lines.
  • If a1/a2 = b1/b2 ≠ c1/c2 , the system will have no solution and will be consistent. The graphs of this type of equation will have parallel lines.
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