Solve Linear Systems Algebraically Part I Chapter 3.2.
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Transcript of Solve Linear Systems Algebraically Part I Chapter 3.2.
![Page 1: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/1.jpg)
Solve Linear Systems Algebraically Part I
Chapter 3.2
![Page 2: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/2.jpg)
Solutions of Linear Systems of Equations
• A linear system of equations will always have one of the following as a solution• Exactly one solution in x and y (the lines intersect in a single point)• An infinite number of solutions (the lines coincide and share all points)• No solution (the lines are parallel and never intersect)
• The next slide shows how graphs of the last two would look
![Page 3: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/3.jpg)
Solutions of Linear Systems of Equations
![Page 4: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/4.jpg)
Solve Linear Systems Algebraically
• Although it is possible to solve a linear system of equations by graphing, this is seldom the best method
• The reason is that, if the solution is not an ordered pair with integer coordinates, then the point of intersection has fractional values
• These are usually impossible to read unless the coordinate plane is broken in the right fractional values
• The best method for solving a linear system of equations is by algebraic methods
![Page 5: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/5.jpg)
Solve Linear Systems Algebraically
• You will learn about two such methods
• The first is called the substitution method
• The second is called the elimination method (or sometimes it is called the addition method)
• In today’s lesson you will solve linear systems by the substitution method
• This method is best used when one or both equations are solved for either y or for x
![Page 6: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/6.jpg)
The Substitution Method
• Suppose you are to solve a linear system of equations like the one below
• Since the solution is the point that is common to both lines, then the x and y values from the first equation must be the same as the x and y values from the second equation
• This means that we can substitute the right part of the first equation into y in the second equation
![Page 7: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/7.jpg)
The Substitution Method
Substitute this:
here:
and solve for x
![Page 8: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/8.jpg)
The Substitution Method
So we have part of the solution:
We need to find y to complete the solution. Do this by substituting for x in the first equation.
![Page 9: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/9.jpg)
The Substitution Method
The solution is
![Page 10: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/10.jpg)
The Substitution Method
• Some linear systems might have both equations solved for y, like the one shown below
• The substitution method is the same: replace y in either equation with the right side of the other equation
![Page 11: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/11.jpg)
The Substitution Method
Substitute this:
here:
and solve for x.
![Page 12: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/12.jpg)
The Substitution Method
So we have part of the solution:
To find y, substitute this value into either equation.
![Page 13: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/13.jpg)
The Substitution Method
The solution is .
![Page 14: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/14.jpg)
A System With No Solution
• How would you know when a system of linear equations has no solution?
• The following example shows what to look for
Use the substitution method
![Page 15: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/15.jpg)
A System With No Solution
• How would you know when a system of linear equations has no solution?
• The following example shows what to look for
• You should get something like , or possibly some other equation that is false
• When this happens, you conclude that the system has no solution
![Page 16: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/16.jpg)
A System With Infinite Solutions
• How do you know when a system has an infinite number of solutions?
• The next example illustrates
• Use the substitution method
![Page 17: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/17.jpg)
A System With Infinite Solutions
• How do you know when a system has an infinite number of solutions?
• The next example illustrates
• You should get something like or possibly
• Both of these are always true, so the system has an infinite number of solutions
![Page 18: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/18.jpg)
Guided Practice
Solve the following systems of linear equations by the substitution method.
1.
2.
3.
4.
![Page 19: Solve Linear Systems Algebraically Part I Chapter 3.2.](https://reader036.fdocuments.in/reader036/viewer/2022062519/5697c02c1a28abf838cd9073/html5/thumbnails/19.jpg)
Exercise 3.2a
• Handout