1. Warm Up!!! Solve the following problems: 1. 4 𝑥+3 =15 2. −4𝑥+6𝑦=4
𝑥+3 =15
2. −4𝑥+6𝑦=4
2𝑥−9𝑦=22
−2𝑥+5𝑦=14
−𝑥+𝑦=1

2. Section 5.1.2
Solving Systems

3. Learning Targets Extraneous Solutions
How to solve systems of equations
How to solve systems of equations using technology

4. Non-Linear Example Solve: 𝒙 +𝟏𝟐=𝟎 𝒙 =−𝟏𝟐 𝒙=𝟏𝟒𝟒
Review the following problem and solution:
Solve: 𝒙 +𝟏𝟐=𝟎
Solution: start by subtracting 12 from both sides:
𝒙 =−𝟏𝟐
Square both sides:
𝒙=𝟏𝟒𝟒

5. Check your answer!!!
Reviewing all the steps, there doesn’t appear to be any obvious mistakes.
However when 𝒙=𝟏𝟒𝟒 is substituted back into original equation:
𝟏𝟒𝟒 +𝟏𝟐=𝟎
𝟏𝟐+𝟏𝟐≠𝟎
WHAT?!?!

6. Vocabulary Extraneous solution:
Extraneous solution is a solution of the simplified form of an equation that does not satisfy the original equation.
Watch out for extraneous solutions, they show up when the variable is under a radical sign or when the variable is in the denominator of a fraction.

7. Systems of Equations
A system of equations is a collection of two or more equations with a same set of variables.
In solving a system of equations, we try to find values for each of the variables that will satisfy every equation in the system.
The equations in the system can be linear or non- linear.

8. Systems of Equations
You are used to solving linear systems of equations.
There are three common methods used to solve systems of linear equations. They are, in no particular order:
Elimination
Equal Value Method (Substitution)
Graphing

9. Practice - Elimination
4𝑥+2𝑦=13 3𝑥+4𝑦=20

10. Practice – Equal Value Method (Substitution)
𝑦= 3 4 𝑥−13 3𝑥+4𝑦=20

11. Non-Linear Systems 𝒇 𝒙 = 𝟐𝒙+𝟑 𝒈 𝒙 =𝒙
Use Substitution 2𝑥+3 =𝑥 Square both sides
2𝑥+3=𝑥 Rearrange all terms
𝑥2− 2𝑥−3=0 Use Quadratic Formula
𝒙=−𝟏 𝑶𝑹 𝒙=𝟑

12. Check answer by graphing:
If there is a square root symbol in original problem, check for extraneous solution
As graph indicates, there is only one point of intersection, where x=3.
Therefore x= -1 is an extraneous solution and should not be counted.

13. *****#5-14 page 224
Solve:
𝒚 𝟏 =𝟐 𝒙 𝟐 +𝟓𝒙−𝟑
𝒚 𝟐 = 𝒙 𝟐 +𝟒𝒙+𝟑

14. 𝒙 𝟐 +𝒙−𝟔=𝟎 rearrange all terms
*****#5-14 page 224
Solve:
𝟐 𝒙 𝟐 +𝟓𝒙−𝟑= 𝒙 𝟐 +𝟒𝒙+𝟑
𝒙 𝟐 +𝒙−𝟔=𝟎 rearrange all terms
Solve for x by quadratic formula:
𝒙=𝟐 𝑶𝑹 𝒙=−𝟑

15. Graphing a system of equations
𝑦 1 = 2𝑥 2 +5𝑥−3 𝑦 2 = 𝑥 2 +4𝑥+3

16. 𝑦= 𝑥 2 +𝑥−6

17. #5-15 Modified!
Solve:
(𝒙−𝟐) 𝟐 +𝟓= 𝟑 𝒙
For now you won’t be able to solve this problem algebraically. Follow the instructions on the next slide to use the “intersect” key on your calculator.

18. To accurately find the coordinates of the point where two functions intersect, perform the following steps:
Graph the functions in a viewing window that contains the point of intersection of the functions.
Press [2nd][TRACE] to access the Calculate menu.
Press [5] to select the intersect option.

19. The next slide summarizes all the steps
Select the first function.
If the name of one of the intersecting functions does not appear at the top of the screen, repeatedly press
Select the second function.
If the calculator does not automatically display the name of the second intersecting function at the top of the screen, repeatedly press arrow key.
The next slide summarizes all the steps

20. Press ENTER Key.
Look at the top left hand corner of the calculator screen to ensure movement between y1 and y2.

21. #5-16 Consider: 𝑓 𝑥 = 12 𝑥 and 𝑔 𝑥 =−(𝑥−3 ) 2 +4
Use your knowledge of parent graphs, to sketch f(x) and graph g(x) on the same set axes.
*****How many times do you think the two functions will intersect?
*****Find all solutions that satisfy both functions. Use Intersect key on your calculator.
Solutions (3,4) (4,3) and (-1,-12)

22. Now use the INTERSECT key to find the answers:
There might be more solutions in the third quadrant.

23. Finding other possible solutions.
In summary, there are three points of intersection:
(3,4) (4,3) and (-1,12)

24. Check your answer using the “store” feature on your calculator
Practice
Graph and solve the following system:
𝑓 𝑥 = 𝑥+3 −4
𝑔 𝑥 = − 𝑥−5 3
Check your answer using the “store” feature on your calculator

25. Graph

26. On your own: Review your notes. Rewrite and fortify them if needed.
Update your vocab list, if needed.
Review and Preview
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