Solving Simultaneous Equations by Substitution

In this lecture, we cover solving simultaneous equations using the method of substitution. This method is often quicker than elimination, particularly when a variable is already isolated.

Key Steps in Substitution Method

1. Rearranging Equations

   • Rearrange one of the equations to isolate a variable.
   • Example: If equation is already in form like x + something, it's ideal for substitution.

2. Substituting into the Other Equation

   • Substitute the isolated variable's expression into the other equation.
   • This eliminates one variable, allowing you to solve for the other.

3. Solving the Equation

   • Simplify and solve the equation that now only contains one variable.
   • Example:
     • Original: 4x - 7y = 6
     • Substitute to get: 4(11 - 3y) - 7y = 6
     • Solve for y: Simplify and find y = 2.

4. Finding the Corresponding Value

   • Substitute back to find the other variable.
   • Example: Use y = 2 in x = 11 - 3y to find x = 5.

5. Verification

   • Optionally, substitute both x and y back into the original equations to verify the solution.

Example 1: Solving a Pair of Equations

• Equations:
  1. 4x - 7y = 6
  2. x = 11 - 3y
• Steps:
  • Substitute x in Equation 2 into Equation 1: 4(11 - 3y) - 7y = 6
  • Simplify to solve for y: y = 2
  • Substitute y = 2 back to find x: x = 5
• Solution: x = 5, y = 2

Example 2: Another Set of Equations

• Equations:
  1. 3x - y = 7
  2. 10x + 3y = -2
• Steps:
  • Isolate y in Equation 1: y = 3x - 7
  • Substitute y in Equation 2: 10x + 3(3x - 7) = -2
  • Simplify to solve for x: x = 1
  • Substitute x = 1 back to find y: y = -4
• Solution: x = 1, y = -4

Notes:

• The substitution method eliminates one of the variables by expressing one variable in terms of the other.
• It’s particularly useful when one of the equations is easily rearranged to isolate a variable.
• Verification by substituting back into the original equations ensures the solution is correct.