Solving Linear Systems
Algebraically
2 Methods for Solving Algebraically
1. Substitution Method
(used mostly when one of the equations has
a variable with a coefficient of 1 or -1)
2. Linear Combination Method
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).
Ex: Solve using substitution method
3x-y=13
2x+2y= -10
1. Solve the 1st eqn for y.
3x-y=13
-y= -3x+13
y=3x-13
2. Now substitute 3x-13 in
for the y in the 2nd
equation.
2x+2(3x-13)= -10
Now, solve for x.
2x+6x-26= -10
8x=16
x=2
3. Now substitute the 2 in
for x in for the equation
from step 1.
y=3(2)-13
y=6-13
y=-7
4. Solution: (2,-7)
Plug in to check soln.
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).
Ex: Solve using lin. combo. method.
2x-6y=19
-3x+2y=10
1. Multiply the entire 2nd
eqn. by 3 so that the
y’s will cancel.
2x-6y=19
-9x+6y=30
2. Now add the 2
equations.
-7x=49
and solve for the
variable.
x=-7
3. Substitute the -7 in for x
in one of the original
equations.
2(-7)-6y=19
-14-6y=19
-6y=33
y= -11/2
4. Now write as an
ordered pair.
(-7, -11/2)
Plug into both equations to
check.
Ex: Solve using either 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.
Ex: Solve using either method.
6x-4y=14
-3x+2y=7
Which method?
Linear combo!
Multiply 2nd eqn by 2.
6x-4y=14
-6x+4y=14
Add together.
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.
Assignment