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A system of equations does not always consist of only linear equations. The aim of this lesson is to show different ways to solve a system of equations if there are quadratic equations in the system.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Finding Intersections with a Circle
Consider the equation of a circle centered at the origin with a radius of
b At what points does the parabola intersect the circle?
Discussion
Nonlinear System
A nonlinear system is a system of equations in which at least one of the equations is nonlinear. Consider the following system of equations.
Even if the first equation in the system is linear, the system is a nonlinear system because the term of Equation (II) is elevated to the power of making it nonlinear.
Pop Quiz
Is the System Nonlinear?
Determine if the given system of equations is nonlinear or linear.
Illustration
The Solutions of a Linear-Quadratic System
Some common nonlinear systems are linear-quadratic systems, which are systems of two equations that consist of a linear equation and a quadratic equation. These systems can have zero, one, or two different solutions. These solutions are the point of intersection of the graphs.
What happens if both equations in a system of two equations are quadratic equations?
Discussion
System of Quadratic Equations
A system of equations does not necessarily consist of linear equations. Different types of equations can be grouped into a system.
A system of quadratic equations is a system of equations that consists of only quadratic equations.
Similar to systems of linear equations, the solution to a system of quadratic equations are the values of the variables that make all the equations true. In the example above, and are a solution to the system. This can be verified by substituting the values into each equation.
In the examples, since the equations remain true, the values are a solution of the quadratic system.Quadratic systems can be solved graphically or algebraically. Since the equations in a quadratic system are not linear, these are nonlinear systems.
Pop Quiz
Is the System Quadratic?
For every given system of equations shown in the applet, determine if the system is a system of quadratic equations or not. Note that every equation in such a system needs to be a quadratic equation.
Illustration
The Solutions of a Quadratic-Quadratic System
Another type of non-linear system is a quadratic-quadratic system, which consists of two quadratic equations. These systems can have zero, one, two, or infinitely many different solutions. These solutions are the points of intersection of the graphs.
Discussion
Solving Nonlinear Systems Graphically
Graphing the equations of a system of equations is helpful. Sometimes it is even possible to solve the system by referencing the graph only.
If the equations can be graphed, the solutions of a nonlinear system are the points of intersection of the graphs of every equation that makes the system. Therefore, by finding the points of intersection of the graphs, it is possible to solve a nonlinear system. As an example, consider a linear-quadratic system.
To find the solutions, first each equation is graphed.
1
Graphing the First Equation
expand_more The first equation of the nonlinear system is a linear equation written in slope-intercept form. By using the slope of and the intercept of the equation is graphed in a coordinate plane.
Then, the second equation needs to be graphed.
2
Graphing the Second Equation
expand_more The second equation is a quadratic equation written in standard form.
To graph a quadratic equation in standard form, first the axis of symmetry needs to be identified and graphed. To do so, the values are substituted into the formula.
This indicates that the axis of symmetry is a vertical line on 
Then, to find the vertex of the parabola, the equation is evaluated to find the value of when
This means that the vertex lies on point 
▼
Solve for
Then, the intercept is the value of of the equation, which in this case is Since the axis of symmetry divides the graph into two mirror images, the parabola also goes through point in
Using these points, it is possible to draw the parabola.
3
Finding the Points of Intersection
expand_more Finally, finding the points in which the graph intersect, the solutions of the nonlinear system are found.
This means that the points and are the solutions of the nonlinear system.
Example
Graphing a Linear-Quadratic System
Emily received a scholarship this summer and attended a camp for skydiving. She has always dreamed of feeling like flying in the sky. Surprisingly, her first task given by her instructor Sky Flyer is to solve a nonlinear system graphically.
Answer
Hint
Graph both equations and find the points of intersection.
Solution
To solve this system graphically, the points of intersection of the graphs need to be found. To do so, each equation should be graphed. Equation (I) is a linear equation that can be written in slope-intercept form.
The slope of and the intercept of can be used to graph this line in the coordinate plane. 
Equation (II) is a quadratic equation written in standard form.
The first step to graph this equation is to find the axis of symmetry. That can be done by substituting the values of and into the Quadratic Formula to find the axis. Note that the root term is
This indicates that the axis of symmetry is the vertical line 
Then, to find the vertex of the parabola, will be substituted for in Equation (II).
This indicates that the vertex is at point 
▼
Simplify
The value of on a quadratic equation written in standard form indicates the value of the intercept. In this case, the value of the intercept is Since the axis of symmetry divides the graph into mirror images, the points and can be added.
These points can be used to graph the parabola.
Looking at the graph, the points of intersection can be located.
The solutions of the nonlinear system are the points and
Example
Graphing a Quadratic-Quadratic System
Instructor Sky Flyer continues teaching Emily about nonlinear systems. Emily is unsure but thinks they must be getting closer to applying this knowledge to something spectacular.
Error loading image.
Sky Flyer gives the following quadratic-quadratic system to Emily and the other students learning to skydive. Answer
Hint
How do you find the axis of symmetry of a parabola if the quadratic equation is written in standard form?
Solution
To solve a quadratic-quadratic system graphically, both quadratic equations are graphed to identify the points of intersection. Looking at the system, it can be noted that both equations of the system are written in standard form.
▼
Solve for
It is possible to find the intercept of the parabola with the value of In Equation (I), this value is Also, since the axis of symmetry mirrors the image of the parabola, there is another known point at These points can be used to graph the parabola.
The value on Equation (II) indicates that the intercept of the second parabola is Since the axis of symmetry mirrors the image, there is another known point
Now that the two equations are graphed, the solutions of the nonlinear system can be determined. They are the points of intersection of the graphs.
The points and are the solutions of the nonlinear system.
Discussion
Solving Nonlinear Systems Using Elimination
Just like in a system of linear equations, there are many methods that can be used to solve a nonlinear system.
Some nonlinear systems can be solved by eliminating one of the variables of the equations involved, similar to the elimination method used in systems of linear equations. Consider the following example.
Since Equation (I) is a quadratic equation, the system is nonlinear. The system can be solved by elimination following the steps below.
It should be noted that not every nonlinear system can be solved using this method.
1
Find a Variable to Eliminate
expand_more There are two different variables in the nonlinear system, and The equations have different powers for the terms of as one is quadratic and the other is linear. An the other hand, both equations have linear terms of Therefore, variable will be eliminated.
2
Eliminating the Variable
expand_more For a variable to be eliminated, its coefficient has to be the same in both equations. This can be achieved by dividing Equation (I) by and then subtracting Equation (II) from Equation (I).
It can be noted that one of the equations only depends on now.
3
Solving the Single Variable Equation
expand_more Equation (II) is a quadratic equation in terms of The equation is already written in standard form.
This equation can be solved using the quadratic formula.
There are two possible values for These can be found by adding or subtracting
UseQuadForm
Use the Quadratic Formula:
CalcPow
Calculate power
Multiply
Multiply
AddTerms
Add terms
CalcRoot
Calculate root
Therefore, the values of that solve the nonlinear system are and
4
Substituting the Values
expand_more Now that there are two known values for the solution values for can be found by substituting the values of into either equation. Since Equation (II) is linear, it is easier to do so in this equation.
Combining each value value with its correspondent value, it can be seen that the solutions can be written as the points and
Example
Skydiving
Instructor Sky Flyer takes the students out to the yard. Emily, super excited, looks up and sees someone free falling with a parachute! 
The height above the ground of the skydiver can be modeled by a quadratic equation.
In this equation, is the time passed in seconds after the jump and is the skydiver's height above the ground in meters. After releasing her parachute, the rate at which she is falling slows. The model for that height is given by the following linear equation.
How long after jumping did the jumper release their parachute? Use the elimination method to find the solution, and round that time to one decimal place.
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Hint
Which variable is the easiest to eliminate?
Solution
First of all, the given equations can be combined to make a nonlinear system.
Finding how long after jumping did the jumper release his parachute is the same as finding the time where the model changed from Equation (I) to Equation (II). This can be done by solving the nonlinear system. The first step to solve the nonlinear system by elimination is to find a variable to eliminate.
The variable is chosen because there are only linear terms of that variable. Then, the elimination is done by subtracting Equation (II) from Equation (I).
Now a quadratic equation was obtained. The equation is already written in standard form.
This equation is solved using the quadratic formula.
The sign indicates that there are two possible values for These can be found by either adding or subtracting the value of the root.
The two possible values for are about seconds and about seconds. Only positive numbers make sense in this case since the skydiver cannot wait a negative time before releasing the parachute. This means that the jumper waited about seconds to release the parachute.
Example
Acrobats on the Circus
Before having the chance to skydive, Instructor Sky Flyer wants to show how the students can train by practicing acrobatics. They all walk into a large gym to see a few pros practice. One acrobat jumped from a platform, and another dove from another higher platform just a split second later. Mid-air, they did a high-five while spinning!
a How long after jumping from their platforms did the acrobats high-five? Solve using elimination.
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b At what height above the floor were the acrobats when they did the high-five?
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Solution
a The first step to solve the given system using elimination is to find a variable to eliminate. The objective is to obtain a single equation that depends on a single variable. The variable is present in only one term, so it can be eliminated to obtain an equation in terms of
b To find the height above the floor at which the acrobats did the high-five, the time found in Part A needs to be substituted into either equation of the given system. In this case, Equation (I) will be used.
Discussion
Solving Nonlinear Systems Using Substitution
In a similar way that a system of linear equations can be solved using the substitution method, there are nonlinear systems that can be solved by substituting. As an example, consider the following linear-quadratic system.
This system is solved using substitution following the steps below.
1
Finding a Term to Substitute
expand_more The first step to solve a nonlinear system using substitution is to identify which term is substituted from one equation to the other. The given system has the variable already isolated so it is easy to select that term.
2
Substituting a Term
expand_more Now the value of from Equation (II) is substituted into Equation (I). Then, the resulting equation is simplified as much as possible.
A quadratic equation that only depends on was obtained.
3
Solving the Resulting Equation
expand_more A quadratic equation was obtained from the previous step. This equation is written in standard form.
These values can be substituted into the quadratic formula.
There are two possible values for depending if there is a subtraction or an addition. These values are calculated individually.
SubstituteValues
Substitute values
CalcPow
Calculate power
Multiply
Multiply
AddTerms
Add terms
CalcRoot
Calculate root
4
Substituting the Values
expand_more To find the values of the values of and are substituted into either equation of the system. It is easier to substitute the values into Equation (II) because it is a linear equation, instead of quadratic.
This indicates that the solutions of the system are the points and
Example
Skydive Recording
Finally, all the math and practice is making sense. Emily is now ready for her first jump! Instructor Sky Flyer prepared an equation to record Emily's height above the ground.
Error loading image.
Her height from the ground when she jumps from the plane, measured in feet, is given by the following quadratic equation.
a After how many seconds does Emily release her parachute? Use substitution to find the answer. Round the time to two decimal places.
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b At what height does Emily release her parachute? Round the height to two decimal places.
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Hint
a Which variable is most convenient to isolate?
b Substitute the answer from Part A into either equation.
Solution
a When Emily releases her parachute, both heights have to be equal. Therefore, to find the time is the same as finding the solution of the following nonlinear system.
The time after Emily jumps has to be positive. Because of this, the only solution that makes sense is the positive one. Therefore, Emily released her parachute about seconds after she jumped.
b To find the height at which Emily released her parachute, the value from Part A can be substituted into either equation. In this case, the value will be substituted into Equation (II) because it is linear. Since the value of is rounded, the equality is substituted with an approximation symbol.
Example
Planning New Formations
As Emily is on cloud nine after skydiving, she dreams of making a formation with other skydivers, where the skydivers make two intersecting circles.
She then realizes that Instructor Sky Flyer would tell her to solve a nonlinear system.
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Hint
Isolate a term of one equation to substitute into the other one.
Solution
To find the points of intersection is the same as solving the system. Neither equation of the system is linear, so this is a nonlinear system. The first step to solve this system using substitutions is to find a variable term that can be isolated. In this case, the convenient term to isolate is
This term is isolated in Equation (II) to be substituted into Equation (I).
The quadratic equation obtained only depends on To solve this equation, first the parenthesis need to be expanded.
This indicates that the circles intersect at To find the coordinate of the intersection points, the value has to be substituted into either equation. The equation when was isolated can be used.
Since the only square root of is the only point of intersection is
▼
Solve for
Closure
Finding the Intersections with a Circle
In this lesson's challenge, an equation of a circle centered at the origin with a radius of was given.
The challenge was to find the points of intersection of that circle.
The term can be isolated on both equations.
The Equation (II) can be subtracted from Equation (I) to solve the system using elimination.
A quadratic equation of is obtained. It can be noted that the equation is written in standard form.
This equation can be solved using the quadratic formula.
Now the two possible values for can be found by adding or subtracting.
Now two values of were obtained, and it should be noted that the value of in Equation (II) has to be greater than or equal to Therefore, the valid solution is while is an extraneous solution. The value of can be substituted into either equation to find the possible values of In this case, it will be substituted into Equation (II).
Here, can be either or because the value of is not limited by any equation. Therefore, the points of intersection are and
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b Given the parabola determine at what points it intersects with the circle. Round the answer to one decimal place if needed.
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Hint
a Consider using substitution.
b Isolate the terms on both equations.
Solution
a The method used to find the points of intersection of the circle and the line is the same as finding the solution of the nonlinear system made by combining these equations.
Finally, these values are substituted into Equation (II) to find the values of of the points of intersection.
The points of intersection are and
b Finding the points of intersection between the circle and a parabola is achieved by finding the solutions of the nonlinear system.
SubstituteValues
Substitute values
CalcPow
Calculate power
Multiply
Multiply
AddTerms
Add terms
CalcRoot
Calculate root
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