1. 3.2 Solving Linear Systems Algebraically
Algebra II

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 or Elimination Method
(use any time)

3. Substitution Method
Solve one of the given equations for one of the variables with a coefficient of 1 or -1.
Substitute the expression from step 1 into the other equation and solve for the remaining variable.
Substitute the value from step 2 into the revised equation from step 1 and solve for the 2nd variable.
Write the solution as an ordered pair (x,y).

4. Ex. 1: Solve using sub. method
Now, solve for x.
2x+6x-26= -10
8x=16
x=2
Now substitute the 2 in for x in for the equation from step 1.
y=3(2)-13
y=6-13
y=-7
Solution: (2,-7)
Plug in to check soln.
3x-y=13
2x+2y= -10
Solve the 1st eqn for y.
-y= -3x+13
y=3x-13
Now substitute 3x-13 in for the y in the 2nd equation.
2x+2(3x-13)= -10

5. 3-2B Solving linear Systems using the elimination method

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

7. Ex. 1: Solve using lin. combo. (elimination) method.
2x-6y=19
-3x+2y=10
Multiply the entire 2nd eqn. by 3.
2x-6y= x– 6y=19
3(-3x+2y=10) -9x+6y=30
-7x=49
2. Now add the 2 eq.’s together
and solve for the variable.
x=-7
Substitute the -7 in for x in one of the original equations.
2(-7)-6y=19
-14-6y=19
-6y=33
y= -11/2
Now write as an ordered pair.
(-7, -5½ )
Plug into both equations to check.

8. Means any point on the line is a solution.
Ex. 2: Choose a method.
9x-3y=15
-3x+y= -5
Which method?
Substitution!
Solve 2nd eqn for y.
y=3x-5
9x-3(3x-5)=15
9x-9x+15=15
15=15
OK, so?
What does this mean?
Both equations are for the same line!
many solutions
Means any point on the line is a solution.

9. Ex. 3: Solve using either method.
6x-4y=14
-3x+2y=7
Which method?
Linear combo!
Multiply 2nd eqn by 2.
+-6x+4y=14
0=28
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.

10. Assignment