Download 3.2.1 * Solving Systems by Combinations

3.2.1 – Solving Systems by
Combinations
• We have addressed the case of using
substitution with linear systems
• When would substitution not be easy to use?
Combinations
• Similar to substitution, we can use a new
method when solving for a specific variable
may not be easy
– Fractions
– Multi-step
– Odd Numbers
• In order to use combinations, our goal is the
following;
• “Knock out” or eliminate one variable. Solve
for the remaining. Then, similar to
substitution, go back and find the other
missing variable
How to use
• To use the combination, or knock-out method, we do the
following
• 1) Find the variable with the same coefficient in both
equations; multiply to get the same coefficient if necessary
• 2) Add or subtract down, make sure one variable is
eliminated
• 3) solve for the remaining variable
• 4) Go back to one of the original equations, and solve for
the remaining variable
• 5) Check final solutions
• To help, it’s generally easiest to line the
equations up as if you were doing addition or
subtraction like you first learned
• Add = if signs are opposite
• Subtract = if signs are same
• Example. Solve the following system.
• 4x – 6y = 24
• 4x – 5y = 8
• Example. Solve the following system.
• 2x – 8y = 10
• -2x – y = -1
• Example. Solve the following system.
• 3x – y = -3
• x+y=3
Multiplying
• As mentioned, sometimes the coefficients
may not be the same
• Allowed to multiply one, or both equations, by
a number to get the same coefficients for one
of the variables
– Make sure to multiple every term!
• Example. Solve the following system.
• 3x + 2y = -2
• x – y = 11
– Which variable should we try to cancel?
• Example. Solve the following system.
• 5x – 2y = -2
• 3x + 5y = 36
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Assignment
Pg. 142
2, 4-6, 9-25 odd
Pg. 143
38, 39