Download Advanced Algebra Notes Chapter 3 – Systems of Linear Equations

Advanced Algebra Notes
Chapter 3 – Systems of Linear Equations and Inequalities
Section 3.1 Solving Linear Systems by Graphing
Objective:
 Graph and solve systems of linear equations in two variables
The solution of a system of linear equations
 an ordered pair (x, y) that satisfies each equation
Possible
1.
2.
3.
Results
Lines intersect  one solution (x, y)
Lines are parallel  No solution
Lines coincide  Infinite solutions ( {(x,y)| - equation }
Examples:
a. x + y = 4
2x + 3y = 9
c. 2y + 3x = 6
4y + 6x = 12
b. 3x + y = -2
6x + 2y = 10
d.
1
x+
2
x–
1
y=2
3
y = -1
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Section 3.2 Solving Linear Systems Algebraically
Objective:
 Use algebraic methods to solve linear systems
 Use algebraic methods to solve real-life problems
Substitution Method
1. Solve for variable (x or y)
2. Substitute that expression in for the 2nd variable – Solve
3. Substitute that value into the original equation and solve
Examples:
a. y = 3x
x + 21 = -2y
b. 4x – y = 11
2x + 2y = 18
c. 5x + 5y = 15
2x + 3y = -1
Linear Combinations – Elimination Method
1. If necessary, multiply one or both equations by a value to make opposite
coefficients on one variable.
2. Add the revised equations. One variable should be eliminated
3. Substitute the value into one of the original equations and solve for the
remaining variable.
4. If both variables are eliminated
a. Resulting equation is true  Infinite Solutions (Identity)
b. Resulting equation is false  No Solution
Examples:
a. 5x + 3y = -1
4x – 3y = -17
d. 5m + 2n = -8
4m + 3n = 2
b. 2x + 3y = -1
-5x + 5y = 15
c. 5x + 4y = 12
7x – 6y = 40
e. x – 2y = 3
2x – 4y = 7
f.
6x – 10y = 12
-15x + 25y = -30
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Applications – Word Problems
Examples:
1. The sum of two numbers is 6. Twice the first number is 9 more than the
second number. Find the numbers
2. Money
A collection of nickels and dimes contains 28 coins. The total value is $1.65.
How many of each coin are there?
3. Digits
The units digit is 4 more than the tens digit. The sum of the original number
plus the number with the digits reversed is 110. Find the number.
4. The sum of the digits of a two-digit number is 6. If 36 is subtracted from the
original number, the result is the number with the digits reversed. Find the
original number
5. Flying with the wind an airplane can travel 3625 km in 6.25 hours, but flying
against the wind the airplane requires 1 hour longer to make the return flight.
Find the air speed of the plane and the speed of the wind.
6. An isosceles triangle has a perimeter of 8.5 cm. A second isosceles triangle,
whose base is equal to that of the first but whose legs are twice as long as
those of the first, has a perimeter of 15.5 cm. Find the lengths of the sides of
each triangle.
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Section 3.3 Graphing and Solving Systems of Inequalities
Objective:
 Graph a system of inequalities to find the solution to the systems
Steps:
1. Graph each inequality
 Draw the boundary line for each
 Shade (lightly or just with arrows showing direction)
2. Shade the overlapping sections
Examples:
a. x – 2y ≤ 3
y > 3x – 4
b. x ≤ 0
y≥0
x – y ≥ -2
c. x ≥ 0
y> 2x – 1
y ≤ 2x + 3
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Section 3.6 Solving Systems of Linear Equations in Three Variables
Objective:
 Solve systems of linear equations in three variables
Solutions are in the form of (x, y, z)
Steps:
1. Use linear combinations to rewrite the linear system in three variables as a
linear system in two variables. (Eliminate one of the variables in two equations)
2. Solve the new linear system for both of its variables
3. Substitute the values found into one of the original equations and solve for the
remaining variable.
4. If you receive a false statement anywhere during the process the system will
have no solution.
Examples:
a. x + 2y + z = 9
3y – z = -1
3z = 12
b. f + s + t = 256
f–s
=6
f+s
= 164
c. 3x + 2y + 4z = 11
2x – y + 3z = 4
5x – 3y + 5z = -1
d. 4x + 3y + 2z = 34
2x + 4y + 3z = 45
3x + 2y + 4z = 47
e. 2a + b + 2c = 5.95
3a + 3b + 3c = 10.41
a + 2b + 2c = 5.35
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