Solving Linear Systems Algebraically

1
Solving Linear Systems Algebraically
2
2 Methods for Solving Algebraically

• Substitution Method
• (used mostly when one of the equations has a
  variable with a coefficient of 1 or -1)
• Linear Combination Method

3
Substitution Method

1. Solve one of the given equations for one of the
   variables. (whichever is the easiest to solve
   for)
2. Substitute the expression from step 1 into the
   other equation and solve for the remaining
   variable.
3. Substitute the value from step 2 into the revised
   equation from step 1 and solve for the 2nd
   variable.
4. Write the solution as an ordered pair (x,y).

4
Ex Solve using substitution method

• Now, solve for x.
• 2x6x-26 -10
• 8x16
• x2
• Now substitute the 2 in for x in for the equation
  from step 1.
• y3(2)-13
• y6-13
• y-7
• Solution (2,-7)
• Plug in to check soln.

• 3x-y13
• 2x2y -10
• Solve the 1st eqn for y.
• 3x-y13
• -y -3x13
• y3x-13
• Now substitute 3x-13 in for the y in the 2nd
  equation.
• 2x2(3x-13) -10

5
Linear Combination Method

1. Multiply one or both equations by a real number
   so that when the equations are added together one
   variable will cancel out.
2. Add the 2 equations together. Solve for the
   remaining variable.
3. Substitute the value form step 2 into one of the
   original equations and solve for the other
   variable.
4. Write the solution as an ordered pair (x,y).

6
Ex Solve using lin. combo. method.

• 2x-6y19
• -3x2y10
• Multiply the entire 2nd eqn. by 3 so that the ys
  will cancel.
• 2x-6y19
• -9x6y30
• Now add the 2 equations.
• -7x49
• and solve for the variable.
• x-7

• Substitute the -7 in for x in one of the original
  equations.
• 2(-7)-6y19
• -14-6y19
• -6y33
• y -11/2
• Now write as an ordered pair.
• (-7, -11/2)
• Plug into both equations to check.

7
Ex Solve using either method.

• 9x-3y15
• -3xy -5
• Which method?
• Substitution!
• Solve 2nd eqn for y.
• y3x-5
• 9x-3(3x-5)15
• 9x-9x1515
• 1515
• OK, so?
• What does this mean?

• Both equations are for the same line!
• many solutions
• Means any point on the line is a solution.

8
Ex Solve using either method.

• 6x-4y14
• -3x2y7
• Which method?
• Linear combo!
• Multiply 2nd eqn by 2.
• 6x-4y14
• -6x4y14
• Add together.
• 028
• Huh?
• What does this mean?

• It means the 2 lines are parallel.
• No solution
• Since the lines do not intersect, they have no
  points in common.