In this section, two types of situations where substitution is used are explored.

Is one of the equations written with one of the variables on one side of the equation? If so, you can substitute that variables value into the other equation and solve it.

Example: Solve

     b = 3
4a + b = 7

Fill in the blanks below. Click "check your answer" to reveal the answers.

• Since b equals 3, substitute Interactive button. Assistance may be required. _____________ 3 into the second equation.
       b = 3
  4a + b = 7
• Solve for a: _____________. Interactive popup. Assistance may be required.

  Check Your Answer

  4a + 3 – 3 = 7 – 3
  
  4a = 4
  4 4 a = 4 4
  a = 1
  
  
• The solution is the point where a equals ____________ and b equals ____________. Graphically this is the point ____________, where the two linear functions intersect. Interactive popup. Assistance may be required.

  Check Your Answer

  The solution is the point where a equals 1 and b equals 3. Graphically this is the point (1, 3), where the two linear functions intersect.

If one of the variables is not written with one variable on one side of the equation, but it is easy to transform into an equation with one variable on one side, substitution could still be used.

Rewrite the equation, and then substitute it into the other equation.

Example: Solve

2a + 3b = 2
 a – 2b = 8

• Solve the second equation for: Interactive button. Assistance may be required.________________ a
  a – 2b = 8
  a – 2b + 2b = 8 + 2b
  a = 2b + 8
• Substitute the new equation into the first equation for a: ___________________. Interactive popup. Assistance may be required.

  Check Your Answer

  2a + 3b = 2
  2(2b + 8) + 3b = 2
• Simplify:____________. Interactive popup. Assistance may be required.

  Check Your Answer

  2(2b + 8) + 3b = 2
  4b + 16 + 3b = 2
  7b + 16 = 2
  
• Solve for b:__________________. Interactive popup. Assistance may be required.

  Check Your Answer

  7b + 16 = 2
  7b + 16 – 16 = 2 – 16
  7b = -14
  7 7 b = -14 7
  b = -2
  
• Substitute the value for Interactive button. Assistance may be required. _____________ b into the second equation to find the value of a:_________________. Interactive popup. Assistance may be required.

  Check Your Answer

  a – 2b = 8
  a = 2b + 8
  a = 2(– 2) + 8
  a = -4 + 8
  a = 4
  
• Check the first equation by substituting the values in. The system's solution is at the coordinate_________________.Graphically this means the two linear functions intersect at this point. Interactive popup. Assistance may be required.

  Check Your Answer

  (4, -2)
• If a variable in one equation of a system of equations is ___________ on ____________ side of the equation, you can substitute for that variable in the other equation. Interactive popup. Assistance may be required.

  Check Your Answer

  alone on one
• If neither equation has a variable alone on one side, but one equation is ___________ to solve for one variable, solve for the ________ and substitute for that variable in the other equation. Interactive popup. Assistance may be required.

  Check Your Answer

  ....one equation is easy to solve for one variable, solve for the variable....