Solving a Linear System
Linear Combinations
Lesson 7.3
Algebra 2
Solving a Linear System by Linear
Combinations
Step 1: Arrange the equations with like terms in columns.
Step 2: Multiply one or both equations by a number to obtain
coefficients that are opposites for one of the variables.
Step 3: Add the equations from Step 2. Combining like
terms will eliminate one of the variables. Solve for the
remaining variable.
Step 4: Substitute the value obtained in Step 3 into either
one of the original equations and solve for the other
variable.
Step 5: Check the solution in each of the original equations.
Example:
4x + 3y = 16
2x – 3y = 8
Example #2
Consider the following linear system:
3x + 2y = 48
9x – 8y = -24
Example #2
Step 1: Multipling the first equation by -3:
(-3)(3x +2y) = (-3)(48)
This gives you
-9x – 6y = -144
Example #2
Step 2: Add the above equation to the original
second equation and simplify.
-9x – 6y = -144
9x – 8y = -12
-14y = -168
y = 12
Example #2
Step 3: Substitute y = 12 back into either original equation
and solve for x:
3x + 2y = 48
3x + 2(12) = 48
3x + 24 = 48
3x = 24
x=8
So, the solution to the system is (x, y) = (8,12)
Observation
Note that the above
system is equivalent to
the following system.
They are equations of
lines with different
slope and different yintercepts.
There is one unique
solution to this system.
Observation
Here is a plot of the
these equations.
Uniqueness and Consistency


Now we consider "what can go wrong" with
the above method.
Example: Consider the following system.
Try to solve it by the linear combination method.
What Happened?
This is clearly a contradiction. We say the
above system is inconsistent.
To see what's happening, write the system
in slope-intercept form and graph the system.
x + 2y = 3
3x + 6y = 3
Parallel Lines
We see the lines are
parallel (same slope with
different y-intercepts).
What else can go wrong?
Here's a second example of "what can go
wrong" with the above methods.
Example:
x + 2y = 3
3x + 6y = 9
Graph the system on the same coordinate pane.
Undetermined (Coinciding)
This says that any ordered pair (x, y) that
satisfies the first equation will also satisfy
the second equation. If we put the system in
y = mx + b form, we get the same equation!
It’s the Graph of an Equivalent
Equation
Here is a plot:
Summary
Given a system of linear equations, there are three
possibilities:
1. Lines have different slope:
- the system has a unique solution.
2. Lines have same slope and different y-intercepts:
- the system has a no solutions. (Parallel Lines)
3. Lines have same slope and y-intercept:
the system has a infinitely many solutions. We
say it is underdetermined. (Coinciding Lines)
End of Lesson