Send the link below via email or IMCopy
Present to your audienceStart remote presentation
- Invited audience members will follow you as you navigate and present
- People invited to a presentation do not need a Prezi account
- This link expires 10 minutes after you close the presentation
- A maximum of 30 users can follow your presentation
- Learn more about this feature in our knowledge base article
Do you really want to delete this prezi?
Neither you, nor the coeditors you shared it with will be able to recover it again.
Make your likes visible on Facebook?
Connect your Facebook account to Prezi and let your likes appear on your timeline.
You can change this under Settings & Account at any time.
Copy of Unit 7 Linear Systems
Transcript of Copy of Unit 7 Linear Systems
Linear equation in two unknowns x and y is an equation of the form
ax + by = c
where a, b, and c are numbers
ex: 4x + 5y = 0
This has a = 4, b = 5, c = 0
When you have two variables and enough information for two equations, we can create a system of equations
A solution of Linear Systems is a pair of values that satisfy both equations
4x + 5y = 40
x-y = 1
This is a system of two linear equations with solution x = 5, y = 4.
We can also write the solution as (5, 4) Part One: Representing Linear Systems The solutions to a single linear equation are the points on its graph, which is a straight line. For a point to represent a solution to two linear equations, it must lie simultaneously on both of the corresponding lines.
To locate solutions to a system of two equations in two unknowns, plot the graphs, and locate the intersection points (if any). Part Two: Solving Graphically Part Three: Solving Systems using Substitution The method of elimination is an algebraic way of obtaining the exact solution(s) of a system of equations in two unknowns by manipulating the equations in such a way as to eliminate of the variables (x or y).
In other words: We can eliminate one variable by subtracting or adding the two equations Part Four: Solving by Elimination Part 5: Properties of Linear Systems The problem with the graphical approach is that it only gives approximate solutions; locating the exact point of intersection of two lines would require perfect accuracy, which is impossible in practice. We can transform a system of two linear equations into a single equation with one variable Ex: y = 3x + 2 7x – 4y = 7 Step One: Notice how one of the variables is isolated. We can use this information and input it into the other equation, instead of the variable.
y = 3x + 2
7x – 4(3x + 2) = 7 Step Two: Solve the resulting equation.
7x – 4(3x + 2) = 7
7x – 12x – 8 = 7
– 5x– 8 = 7
– 5x = 7 + 8
– 5x = 15
– 5 – 5
x = – 3 Step Three: Input the new value into either of the equations and solve for the missing variable.
7 (-3) – 4y = 7
-21 – 4y = 7
- 4y = 7 + 21
- 4y = 28
- 4 -4
y = - 7 y = 3 (-3) + 2
y = -9 +2
y = -7 Step One: Make sure both equations are in standard form (Ax + By = C) x + 2y = 9
– x + 3y = 16 Step Two: Decide on addition or subtraction so that one of the variables (in our case x) will cancel each other out. Step Three: Now solve for the remaining variable
y = 5 Step Four: Using one of the original equations, input the new value for the variable solved in the earlier steps.
- x + 3(5) = 16
-x + 15 = 16
–x = 16 - 15
-x = 1
x = -1 x + 2 (5) = 9
x + 10 = 9
x = 9 – 10
x = -1 So the solution for the system (x, y) = (-1, 5) TRY:
4x – 3y = 25
–3x + 8y = 10 Answer y = 5 x = 10 The solution is (10,5) There are 3 possibilities of solutions when graphing. Single solution: Where the slopes are different but the y-intercept can be the same. These lines can also be perpendicular, which means they intersect; their lines form a 90-degree angle, they are also negative reciprocals of each other. No Solution: They would have the same slope, but different y-intercepts. When they have no solution the lines are parallel, parallel lines continue, literally, forever without touching. Infinite Solutions: Where they have the same slope and y-intercept. Extra Resources The Math Warehouse http://www.mathwarehouse.com/ Tutorials for Finite Math http://people.hofstra.edu/stefan_waner/realworld/tutorialsf1/frames2_1.html YourTeacher - YouTube Solving Systems of Equations - YourTeacher.com - Algebra Help Solving Systems by Substitution - YourTeacher.com - Math Help Solving Systems by Graphing - YourTeacher.com - Algebra Help TRY: Determine which of the 3 categories each question belongs to Answers 1. 2. 3. 1. 2. 3. TRY:
y = 3x – 2
y = -x + 2 Answer Blue: y = 3x – 2
Red: y = -x + 2 Example y = -x + 5
y = 1 x + 2
2 Purple Math http://www.purplemath.com/modules/systlin4.htm TRY:
2x – 3y = –2
4x + y = 24 Answer