Here are some more examples of using substitution to solve simultaneous equations:

1. 3x + y = 13
   5x -2y = 7
   
   The coefficient of y in Equation 1 is 1. So first we make y the subject of Equation 1:
   
   y = 13 - 3x
   
   Next, substitute this expression for y in Equation 2 and solve for x:
   
   

   5x - 2(13 - 3x) = 7	Multiply out bracket
   5x - 26 + 6x = 7	Combine like terms (x's on one side, numbers on the other)
   11x = 33	Divide both sides by 11 to solve for x
   x = 3

   
   
   Finally, substitute the solution for x into the expression for y:
   
   y = 13 - 3(3) = 4
   y = 4
   
   So the solution to the pair of simultaenous linear equations is (3,4).
   
   
   
2. 2x + 4y = 10
   2x + y = 4
   
   The coefficient of y in Equation 2 is 1. So first we make y the subject of Equation 2:
   
   y = 4 - 2x
   
   Next, substitute this expression for y in Equation 1 and solve for x:
   
   

   2x + 4(4 - 2x) = 10	Multiply out bracket
   2x + 16 - 8x = 10	Combine like terms (x's on one side, numbers on the other)
   -6x = -6	Divide both sides by -6 to solve for x
   x = 1

   
   
   Finally, substitute the solution for x into the expression for y:
   
   y = 4 - 2(1) = 2
   y = 2
   
   So the solution to the pair of simultaenous linear equations is (1,2).
   
   
   
3.   x - 5y = 7
   2x -4y = 8
   
   The coefficient of x in Equation 1 is 1. So first we make x the subject of Equation 1:
   
   x = 7 + 5y
   
   Next, substitute this expression for x in Equation 2 and solve for y:
   
   

   2(7 + 5y) - 4y = 8	Multiply out bracket
   14 + 10y - 4y = 8	Combine like terms (y's on one side, numbers on the other)
   6y = -6	Divide both sides by 6 to solve for y
   y = -1

   
   
   Finally, substitute the solution for y into the expression for x:
   
   x = 7 + 5(-1) = 2
   x = 2
   
   So the solution to the pair of simultaenous linear equations is (2,-1).
   
   
   
4. 2x + 4y = 12
     x + 8y = 30
   
   The coefficient of x in Equation 2 is 1. So first we make x the subject of Equation 2:
   
   x = 30 - 8y
   
   Next, substitute this expression for x in Equation 1 and solve for y:
   
   

   2(30 - 8y) + 4y = 12	Multiply out bracket
   60 - 16y + 4y = 12	Combine like terms (y's on one side, numbers on the other)
   -12y = -48	Divide both sides by -12 to solve for y
   y = 4

   
   
   Finally, substitute the solution for y into the expression for x:
   
   x = 30 - 8(4) = -2
   x = -2
   
   So the solution to the pair of simultaenous linear equations is (-2,2).
   
   
   
5. 2x - 4y = 10
   -4x+5y = -26
   
   None of the coefficients are 1. So we can choose to make any variable the subject.
   Lets make x the subject of Equation 1:
   
   x = (10 + 4y)/2
   x = 5 + 2y
   
   Next, substitute this expression for x in Equation 2 and solve for y:
   
   -4(5 + 2y ) + 5y = -26
   -20 - 8y + 5y = -26
   -3y = -6
   y = 2
   
   Finally, substitute the solution for y into the expression for x:
   
   x = 5 + 2(2) = 9
   x = 9
   
   So the solution to the pair of simultaenous linear equations is (9,2).
   
   
   
6. 6x + 2y = 10
   10x - 3y = 12
   
   None of the coefficients are 1. So we can choose to make any variable the subject.
   Lets make y the subject of Equation 2:
   
   y = (12-10x)/(-3)
   y = -4 + (10/3) x
   
   Next, substitute this expression for y in Equation 1 and solve for x:
   
   6x + 2(-4 + (10/3) x) = 10
   6x - 8 + (20/3) x = 10
   (38/3) x = 18
   x = 18*(3/38) = 27/19
   
   Finally, substitute the solution for x into the expression for y:
   
   y = -4 + (10/3)/(27/19) = -4 + 270/57 = -228/57 = 270/57 = 42/57 = 14/19
   
   So the solution to the pair of simultaenous linear equations is (27/19,5/19).
   
   
   
   

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