1. How Can I use a graph to solve?
Section 5.1.2
How Can I use a graph to solve?

2. Review solve each system
Answers:
( -7,-4)
( 3,4)
( -1,2) B) C)

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

4. Systems of Linear Equations
You are used to solving linear systems of equations.
There are four common methods used to solve Linear Systems of equations. The methods, in no particular order, are:
Elimination
Graphing
Equal Value method
Substitution.

5. Review and Practice Solve: 𝒙 +𝟏𝟐=𝟎
Review the following problem and solution:
Solve: 𝒙 +𝟏𝟐=𝟎
Solution: start by subtracting 12 from both sides:
𝑥 =−12
Square both sides:
( 𝑥 ) 2 = (−12) 2
x=144

6. Check your answer
By reviewing all the steps, there doesn’t seem to be any oblivious mistakes.
However when x=144 is substituted back into original equation: =0
12+12≠0

7. 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 to show when the unknown is under a radical sign or when the unknown is in the denominator of a fraction.

8. Other Systems Consider the following functions. 𝒇 𝒙 = 𝟐𝒙+𝟑 and 𝒈 𝒙 =𝒙
Sketch each graph by hand. How many POIs are possible?
Solve by using any method. Then check your answer using a different method.

9. 𝒇 𝒙 = 𝟐𝒙+𝟑 𝒈 𝒙 =𝒙
Use the equal values method (a form of Substitution) 𝟐𝒙+𝟑 =𝒙 Square both sides 𝟐𝒙+𝟑=𝒙𝟐 Rearrange all terms 𝒙𝟐− 𝟐𝒙−𝟑=𝟎 apply the QF or factor 𝒙=−𝟏 𝑶𝑹 𝒙=𝟑

10. 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.

11. Click here for more examples

12. 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 factoring or using the quadratic formula:
𝒙−𝟐 𝒙+𝟑 =𝟎
𝒙=𝟐 𝑶𝑹 𝒙=−𝟑

13. Solve by graphing
You should be able to identify solutions on a graph also.
The next 2 slides demonstrate how to locate the solution for problem 5-14 on a graph.

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

15. 𝑦= 𝑥 2 +𝑥−6

16. #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.

17. 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.

18. 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

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

20. #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)

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

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

23. 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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# 18-28
Problem 28c: use at least two methods… try this problem before leaving.