Linear equations in two variables:

If a, b, c, are real numbers (a ≠ 0, b ≠ 0) then ax + by = c is an equation in two variables. The equation is said to be satisfied for those values of x and y respectively. When there is just one linear equation containing two variables, then an infinite number of its solutions can be found. Let us consider two linear equations 2x + 3y = 4 and -x + 2y = 12. When we consider two linear equations with two variables, such equations are called Systems of equation or simultaneous equations

Solving Systems of equation by substitution method

Solve the following systems of equations (simultaneous equations) by expressing x in terms of y or y in terms x

  1. 5x+3y = 13 ; y+4x = 9
   5x+3y = 13 -------------------(1)   

   Consider y+4x = 9
                y = 9 - 4x
   Substitute y = 9-4x in equation (1)
   5x+3(9-4x) = 13 
    5x+27-12x = 13 Use distributive property
   -7x + 27   =13 simplify
    -7x = 13-27 subtract both sides by 27
     -7x = -14 
       x = 2 divide both sides by-7
       y = 9-4(2)= 1
       y = 1
Solution (2,1) 
2. 3x-4y = 7; 2x+y = 12
   3x-4y = 7-------------(1)
   2x+y  = 12
       y = 12-2x
   Substitute y= 12-2x in equation (1)
    3x-4(12-2x)=7
      3x-48+8x = 7
        11x-48 = 7
           11x = 55
             x = 5
           y= 12-2(5)
           y = 2
     Solution (5,2)
3. 2x-3y = 14; 5x+4y = 12
   5x+4y = 12----------(2)
   2x = 14+3y
    
x= \frac{14+3y}2

Substitute x value in equation (2)

5(\frac{14+3y}2) +4y = 12
multiply all the terms by 2
2(5(\frac{14+3y}2)+2(4y) = 2(12)
                             5(14+3y)+8y = 24
                               70+15y+8y = 24
                                     23y = 24-70
                                     23y = -46
                                       y = -46/23
                                       y = -2
x= \frac{14+3(-2)}2= \frac{8}2=4
                                    x = 4
                             Solution = (4, -2)

Solving Special Systems of Equations