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Warm Up
1) Find the 43rd term of the sequence 47, 34, 21, 8, ….
2) Rewrite in slope-intercept form -5y + 3x = -9
Homework CheckPage 21a) t = 14c – 4 b) $803a) c = .02n + .30 b) 740 minutes
Page 31)a)4.29x + 3.99y = 30 b) 5.1 pounds4) a) 2x + 5y = 150 b) (0,30) (75,0)
(50,10)
Homework Check 2.2 (Page 4) 1) y = 4/3x – 8 2) y = 1/2x – 7
3) a)2 b) 3 c) 1 d) 4
4) a) y = .75x + 3 b) $9c) 16 miles
5) a) 2W + 1T = 20 b) 6 tie games
Systems of Equations
OBJECTIVES• To understand what a system of
equations is. • Be able to solve a system of
equations from graphing, substitution, or elimination
• Determine whether the system has one solution, no solution, or an infinite amount of solutions.
Defining a System of Equations
• A system of equations is a grouping of 2 or more equations, containing one or more variables.
Examples:
x + y = 2
2x + y = 5
2y = x + 2
y = 5x - 7
6x - y = 5
Solution?
To be a solution to the system, all equations must be satisfied.
Is (-3, 4) a solution to the system?
Types of SolutionsIntersecting Lines have ONE unique solution.
Coincidental Lines (or same lines) have infinitely MANY solutions.
Parallel Lines have NO solutions!
Solving Systems of Equations
• There are three methods to solving a system of equations
1. Graphing2. Substitution3. Elimination
Graphing
Graphing CalculatorStep 1:
Step 2:
Step 3:
Step 4:
Examples…
1)
Determine whether the following have one, none, or infinitely many solutions
2y + x = 8
y = 2x + 4
3) 2) x - 5y = 10
-5y = -x +6
y = -6x + 8
y + 6x = 8
ANS: One Solution
ANS: No Solution
ANS: Infinite Solutions
Solving Systems by Substitution
Solving by Substitution Steps
If one of the equations is already solved for a variable, Substitution may be an easy method to solve.
Step 1: Make sure one of the equations is solved for a variable.
Step 2: Substitute the expression into the other equation.
Step 3: Solve for the variable.
Step 4: Substitute the value of x (or y) into either equation and solve.
ExampleSolve the system by substitution.
Solving Systems by Elimination
Elimination
We can solve by elimination by either Adding or Subtracting two equations to eliminate a variable!
Example
Solve by Elimination
Example
Solve by Elimination
More with Elimination
If one will not cancel, multiply one or both equations to get variables to cancel
Example
Solve the system by elimination.
Special Solutions
Solve each system by elimination.
1. 2.
Homework