Unit 13: Nonlinear Systems of Equations
Where a line, a parabola, or a circle actually cross each other, and how many times they're allowed to.
5 topics · 18 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Solving a System of a Line and a Parabola
- How it worksHow does substituting the linear equation into the quadratic reduce the system to one equation in one variable?
- Worked exampleFind where the line y = x + 1 crosses the parabola y = x2 - 5.
- Common mistakeWhy doesn't every line drawn on a parabola's graph actually intersect it at two real points?
Your turn1 of 2
Find the sum of the x-coordinates where the line y = 7x + 6 crosses the parabola y = x2 - 8.Solving a System with a Circle and a Line
- How it worksHow does substituting a line's equation into a circle's equation turn the system into a single quadratic?
- Worked exampleSolve the system x2 + y2 = 25 and y = x - 1 to find both points where the line pierces the circle.
- Common mistakeWhy doesn't a discriminant of zero in the resulting quadratic mean the line missed the circle?
- Going furtherWhat happens to the system's solutions when the line is tangent to the circle instead of cutting through it?
Your turn1 of 2
Find the sum of the y-coordinates of the two points where the line y = x - 4 intersects the circle x2 + y2 = 75.Solving a System of Two Conics
- How it worksHow does elimination let you cancel the squared terms in a system of two conic equations?
- Worked exampleSolve the system x2 + y2 = 10 and x2 - y2 = 2 for all real solutions.
- Common mistakeWhy doesn't solving a system of two conics algebraically always give you every intersection visible on the graph?
- Ties togetherHow does canceling the squared terms between two conics use the same elimination strategy as solving a linear system?
Your turn1 of 2
Solve the system x2 + y2 = 41 and x2 - y2 = 10 for x2 (add the two equations and solve).Counting the Solutions of a Nonlinear System Graphically
- How it worksHow does sliding a line up and down against a fixed parabola change the number of intersection points?
- Going furtherWhat happens to the number of real solutions of a linear-quadratic system as the line's y-intercept increases?
- Common mistakeWhy can a system of two equations have complex solutions that never show up as crossing points on the real graph?
Your turn1 of 2
A line y = x + b is compared against the fixed parabola y = x2 - 7. Find the value of b at which the line becomes exactly tangent to the parabola.Applications of Nonlinear Systems
- How it worksHow do you set two real-world models, like a parabolic path and a fixed height, equal to find where they meet?
- Worked exampleFind when a ball following h(t) = -16t2 + 40t + 5 reaches a height of 20 feet by solving the resulting system.
- Common mistakeWhy can a real-world nonlinear system produce a mathematical solution that doesn't make physical sense?
- Ties togetherHow does finding when a thrown ball reaches a target height use the same substitution method as solving any nonlinear system?
Your turn1 of 2
A ball follows h(t) = -16t2 + 48t + 13. Find the sum of the two times (in seconds) at which it reaches a height of 24 feet.