General Info
What is the term "system" referring to in a system of linear equations / inequalities?
A set of more than one equation / inequality.
What are the two algebraic methods?
Substitution
Elimination
y = 6x - 7
x = 3
(3, 11)
y = 2x - 1
3x - 2y = 4
(-2, -5)
What is the FIRST (easily overlooked, but quite crucial) step to solving a word problem?
Define your variables.
List three limitations of the Graphing Method.
Time
Possibility of non-integer values
etc
This method uses the Properties of Equality to REPLACE one of the variables with an equivalent expression involving the other variable.
The Substitution Method
x + y = 85
x - y = 33
(59, 26)
What is necessary before using the Substitution Method?
One of the variables must be isolated / have been solved for.
"5 red bricks and 3 green blocks weigh 31 pounds. 2 red bricks and 1 green block weigh 12 pounds. How much does each red brick and green block weigh?"
What would our variables represent in this problem?
The weight of a red brick and a green block.
What civilization was largely responsible for expanding and sharing Algebra with the rest of the world?
Persians
This method uses the Properties of Equality to either add or subtract the two equations together in order to cancel out one of the variable terms.
The Elimination Method
What is necessary before using the Elimination Method?
y = 2x + 8
y = x + 8
(0, 8)
"5 red bricks and 3 green blocks weigh 31 pounds. 2 red bricks and 1 green block weigh 12 pounds. How much does each red brick and green block weigh?"
Write the system of equations.
5x + 3y = 31
2x + y = 12
How many zeroes follow the "1" in a Googol?
100 zeroes
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
This method involves plotting the two equations on a coordinate plane, and looking for their intersection point.
The Graphing Method
3x + y = 14
y = x - 6
(5, -1)
x - 2y = -3
3x + 4y = 21
(3, 3)
"5 red bricks and 3 green blocks weigh 31 pounds. 2 red bricks and 1 green block weigh 12 pounds. How much does each red brick and green block weigh?"
How much would 10 red bricks weigh?
10 red bricks would weigh 50 pounds.
How do we determine the solution set of a system of inequalities?
The section of the coordinate plane covered by overlapping shaded regions.
Of the three methods discussed, this method is the only one that does NOT use the Properties of Equality, and thus is the only viable method to find solutions to systems of inequalities.
The Graphing Method
If you wanted to use the Elimination Method but there were no terms with the same coefficients in both equations, what could/should you do?
Multiply one or both of the equations by a given factor to create a common term.
What would it look like when using the GRAPHING METHOD if a system of equations had NO solutions?
Two parallel lines.
"5 red bricks and 3 green blocks weigh 31 pounds. 2 red bricks and 1 green block weigh 12 pounds. How much does each red brick and green block weigh?"
How much does a green block weigh?
A green block weighs 2 pounds.