a Solve the equation for y. Then substitute the found solutions into the equation to check if they are extraneous solutions or not.
B
b Solve the equation for y. Then substitute the found solutions into the equation to check if they are extraneous solutions or not.
C
c Compare the solution and the extraneous solution from Parts A and B. How do they relate to each other?
A
aSolution: x=9
Extraneous Solution: x=- 2
B
bSolution: x=- 2
Extraneous Solution: x=9
C
c They will switch places. See solution.
Practice makes perfect
a Let's solve the given equation and find the value of y. We can begin by squaring both sides of the equation. Then, we will try to separate y on one side and the rest of the terms on another side.
By the Zero Product Property, at least one of the factors must be equal to 0. Therefore, we get two possible solutions.
ly-9=0 y_1=9 and ly+2=0 y_2=- 2
Now, let's recall that an extraneous solution is an apparent solution that does not satisfy the original equation. To check if either of these solutions is extraneous, we will substitute each of them into the original equation and see if it remains true.
sqrt(7y+18)=y
y_1=9
y_2=- 2
sqrt(7( 9)+18)? = 9
sqrt(7( - 2)+18)? = - 2
sqrt(63+18)? =9
sqrt(- 14+18)? =- 2
9=9
2≠- 2
We can conclude that the solution of the original equation is 9, while - 2 is an extraneous solution.
b As requested, we will begin by multiplying one side of the given equation by - 1. Let's look at what happens when we do this to the right-hand side of the equation.
sqrt(7y+18)=y*( - 1) ⇒ sqrt(7y+18)=- yTo solve this new equation, once again we can start by squaring both sides.
(sqrt(7y+18))^2&=(- y)^2
⇓ &
7y+18&=y^2
When we do this, we get the exact same equation as we had in Part A. Therefore, solving in the same way, we will get solutions y_1=9 and y_2=- 2. Let's substitute each of them into our new equation to see if it remains true.
sqrt(7y+18)=- y
y_1=9
y_2=- 2
sqrt(7( 9)+18)? =- 9
sqrt(7( - 2)+18)? =- ( - 2)
sqrt(63+18)? =- 9
sqrt(- 14+18)? =2
9≠- 9
2=2
The solution of the new equation is - 2. The value 9 is an extraneous solution.
c Let's analyze the solutions and the extraneous solutions of the equations from Parts A and B.
Part A
Part B
Equation
sqrt(7y+18)=y
sqrt(7y+18)=- y
Solution
y= 9
y= - 2
Extraneous Solution
y= - 2
y= 9
As we can see, by multiplying one side of the original equation by - 1, the solution and the extraneous solution switched. Therefore, we can make a conjecture about the similar equation, sqrt(y+2)=y. If we multiply one side of this equation by - 1, the solution and the extraneous solution will switch places.
The solution of the original equation will become an extraneous solution of the new equation.
The extraneous solution of the original equation will become the solution of the new equation.
