+Learning Targets
Extraneous SolutionsHow to solve systems of equationsHow to solve systems of equations
using technology
+Non-Linear Example
Review the following problem and solution:
Solve: Solution: start by subtracting 12 from both sides:
Square both sides:
+Check your answer!!!
Reviewing all the steps, there doesn’t appear to be any obvious mistakes.
However when is substituted back into original equation:
WHAT?!?!
+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.
+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.
+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:
1) Elimination2) Equal Value Method (Substitution)3) Graphing
+Non-Linear SystemsUse Substitution
Square both sides
Rearrange all terms
Use Quadratic Formula
𝒙=−𝟏𝑶𝑹𝒙=𝟑
+Check answer by graphing:
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-5-4-3-2-1
12345
x
y
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.
If there is a square root symbol in original
problem, check for
extraneous solution
+
-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9
-8
-6-4
-2
24
68
1012
1416
x
y
(-3,0)
(2,15)
Graphing a system of equations
+#5-15 Modified!
Solve:
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.
To accurately find the coordinates of the point where two functions intersect, perform the following steps:1. Graph the functions in a viewing window that contains the point of
intersection of the functions.
2. Press [2nd][TRACE] to access the Calculate menu.
3. Press [5] to select the intersect option.
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
Press ENTER Key. Look at the top left hand
corner of the calculator screen to ensure movement between y1 and y2.
+#5-16
Consider: and +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.
Finding other possible solutions.
In summary, there are three points of intersection:(3,4) (4,3) and (-1,12)
+Practice
Graph and solve the following system:
Check your answer using the “store” feature on your calculator
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