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College Algebra Answer/Discussion to PracticeProblems on Solving Systems of NonlinearEquations in Two Variables
Answer/Discussionto 1a
Im going to chose to use the substitution method.
Not that because of the terms involved, it would not be possible towork this problem using the process of elimination.
Step 1: Simplifyif needed.
Both of these equations are already simplified. No work needsto be done here.
Step 2: Solveone equation for either variable.
It does not matter which equation or which variable you choose to solvefor. Just keep it simple.
Im going to chose to use the first equation to solve for y.
Solving the first equation for y we get:
*1st equation solved for y
Step 3: Substitutewhat you get for step 2 into the other equation AND Step 4: Solvefor the remaining variable .
Substitute the expression 4/x foryinto the second equation and solve forx: (when you plug in an expression like this, it is just like you plugin a number for your variable)
*Sub. 4/x in for y
*Factorthe trinomial
*Factorthe difference of squares
*Set 1st factor = 0
*Set 2nd factor = 0
Step 5: Solvefor second variable.
Plug in both values found for x in step4.
Plug in -2 for x into the equation instep 2 to find ys value.
*Plug in -2 for x
(-2, -2) is one solution to this system.
Plug in 2 for x into the equation instep 2 to find ys value.
*Plug in 2 for x
(2, 2) is another solution to this system.
Step 6: Checkthe proposed ordered pair solution(s) in BOTH original equations.
You will find that if you plug either the ordered pair (-2, -2) OR(2, 2) into BOTH equations of the original system, that they are both asolution to BOTH of them.
(-2, -2) and (2, 2) are both a solution to our system.
Answer/Discussionto 1b
Im going to chose to use the elimination by addition method, howeverit would be perfectly find for you to use the substitution method. Either way the answer will be the same.
Step 1: Simplifyif needed.
Both of these equations are already simplified. No work needsto be done here.
Step 2: Multiplyone or both equations by a number that will create opposite coefficientsfor like terms if needed.
Again, you want to make this as simple as possible.
The variable that you want to eliminate must be a like variable. Note that x squareds' coefficients arealready opposites. So we do not have to multiply either equationby a number.
Also note that the y terms are not liketerms so we would not be able to eliminate yin this problem.
Here is the original problem complete with opposite coefficientsfor the x squared terms:
Step 3: Addequations.
*Note that x squaredsdropped out
Step 4: Solvefor remaining variable.
*Setting the quadratic = 0
Note that this does not factor, so Im going to have to use drasticmajors - yes the Quadratic Formula.
If you need a review on the quadratic formula feel free to
*Identifying a,b, and cfor the quad. form.
*Plugging a, b,and c into the quad form.
*Simplifying radicand, which is a negativenumber
Note how we came up with a negative underneath the square root.
That means there is NOT a real number solution for this.
Step 5: Solvefor second variable.
There is no value to plug in here.
Step 6: Checkthe proposed ordered pair solution(s) in BOTH original equations.
There are no ordered pairs to check.
The answer is no solution.
Answer/Discussionto 1c
Im going to chose to use the substitution method, however it wouldbe perfectly find for you to use the elimination by addition method. Either way the answer will be the same.
Step 1: Simplifyif needed.
Both of these equations are already simplified. No work needsto be done here.
Step 2: Solveone equation for either variable.
Note how the first equation is already solved for y. We can use that one for this step.
It does not matter which equation or which variable you choose to solvefor. But it is to your advantage to keep it as simple as possible.
First equation solved for y:
*Solved for y
Step 3: Substitutewhat you get for step 2 into the other equation AND Step 4: Solvefor the remaining variable.
Substitute the expression fory into the second equation and solve forx: (when you plug in an expression like this, it is just like you plugin a number for your variable)
*Sub. x^2 + 1fory *Distribute -2 through ( )
*Variable dropped out AND true
Wait a minute, where did our variable go????
As mentioned before, if the variable drops out AND we have a TRUE statement,then when have an infinite number of solutions. They end up beingthe same line.
Step 5: Solvefor second variable.
Since we did not get a value for x, thereis nothing to plug in here.
Step 6: Checkthe proposed ordered pair solution(s) in BOTH original equations.
There is no value to plug in here.
When they end up being the same equation, you have an infinite numberof solutions. You can write up your answer by writing out eitherequation to indicate that they are the same equation.
Answer/Discussionto 1a
fory into the second equation and solve forx:
OR 
.