3.1 - solving systems by graphing. all i do is solve!
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3.1 - Solving Systems by Graphing
All I do is Solve!
Graph the following pairs of equations:
a. y = x + 5 y = -2x + 5
b. y = 3x + 2 y = 3x - 1
c. y = -4x - 2 y = 8x + 4 -2
System of Equations: A set of two or more equations that use the same variables.
Solution to a System: A set of values that makes ALL equations true.
Is (-3, 4) a solution to the system?
Is (3,1) a solution for the following system?
Types of Solutions
1. Intersecting Lines have ONE unique solution.
2. Coincidental Lines (or same lines) have MANY solutions.
3. Parallel Lines have NO solutions!
Solve by Graphing: y4x 6y2x 10
Graphing Calculator
Solve the following systems by graphing:
A.yx5y 2x5
B.y3x2y3x 1
C.y 2x 2
y 42x 2
Solving by Graphing:x 2y 72x 3y0
Classwork (To Be Turned In):
What type of solution does each system have? If the solution exist, what is it?
2)
3)
1)
4)
3.2 Solving Systems Algebraically
Solving Systems by Substitution
The Substitution Method
• Not every system can be solved easily by graphing. Sometimes it is not always clear from the graph where the solution is.
• We can use an algebraic method called SUBSTITUTION to find the exact solution without a graphing calculator.
Solving by Substitution
1. Solve for one of the variables.
2. Substitute the expression of the equation you solved for into the other equation.
3. Solve for the variable.
4. Substitute the value of x into either equation and solve.
Solve the system by substitution.
You Try! Solve 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!
Solve by Elimination:
Solve by Elimination:
You Try! Solve the systems by elimination:
Note: Sometimes with elimination you will have to multiply one or both of the equations in a system. This creates an EQUIVALENT SYSTEM that has the same solution to the original.
Solve the system by elimination.
Special Solutions
Solve each system by elimination.
1. 2.
3.6 Solving Systems with Three Variables
Systems with 3 variables will have 3 equations. These type of systems are in three dimensions! So it is not going to be easy to find their solution by graphing.
3-variable Systems
We can solve Systems with 3 variables, using Elimination OR Substitution.
Solve by Elimination
Luckily, we have an easier way to do this!
When solving system of the equations we can use Matrices!!
SO MUCH WORK!!!
Writing Systems as a Matrix Equation
For Matrix Equations in the formAX = B
•A is called the COEFFICIENT MATRIX•X is called the VARIABLE MATRIX•B is called the CONSTANT MATRIX
•A Coefficient is a number INFRONT of a variable.
•A Variable is a value represented by a letter or symbol
•A Constant is a number WITHOUT a variable.
Write the following System as a Matrix:
Remember that when solving matrix equations:
If AX=B then X = A-1B
Solve the Matrix Equation
Write as a Matrix Equations and Solve!
Write the follow systems as Matrix Equations. Then Solve!1. 2.
Unique Solutions
Remember, Systems can have 1 solution, NO solutions, or MANY solutions.
IF matrix A’s Determinate is 0 then the matrix does NOT have an inverse and the systems does NOT have a unique solution.
IF matrix A’s Determine is NOT 0 then the matrix has an inverse and the system has a unique solutions!
Unique Solution
Determine if there is a unique solution.
ExampleThe sum of three numbers is 12. The 1st is 5 times the 2nd. The sum of the 1st and 3rd is 9. Find the numbers.
HW 3.6/ Classwork
x 2y z42x y 4z 8 3x y 2z 1
4A 2U I 25A 3U 2I 17A 5U 3
2l 2w h72l 3wh2w
x 2y22x 3y z 94x 2y 5z1
6x y 4z 8y
4z
60
2x z 2
5z 4y43x 2y0x 3z 8
4x y z 5 x y z52x z 1y
26. 27. 28.
29. 30. 31.
32.
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