Algebra 2 Systems of Linear and Quadratic Equations Worksheet (PDF)
This nonlinear systems worksheet for Algebra 2 covers systems of linear and quadratic equations (Common Core A-REI.C.7), with a free PDF to print.
These 20 problems practice systems of linear and quadratic equations (Common Core A-REI.C.7). The first problems check checking points and solving by substitution, the middle problems add tangent lines, circles, and counting intersections with the discriminant, and the last six are word problems and multi-step tasks that need break-even, ball and ramp, and number problems.
Work in order and show every step. Write each answer in simplest form, and open the answer key only after you have tried the problem yourself.
Algebra 2 · Common Core State Standards: A-REI.C.7
Nonlinear systems worksheet: exercises 1 to 10
Exercise 1 – Check a point
Exercise 2 – Substitution
Exercise 3 – Parabola and line
Exercise 4 – Read the graph
The system \(y = x^{2} + 2x – 3\) and \(y = -x + 1\) is graphed. Estimate the intersection points from the graph, then solve algebraically to confirm.
Exercise 5 – Tangent line
Exercise 6 – No real solution
Exercise 7 – Line in standard form
Exercise 8 – Irrational solutions
Exercise 9 – Solve and graph
Solve \(y = x^{2} – 4\) and \(y = -x + 2\), and mark the solutions on a sketch.
Exercise 10 – Circle and line
Exercises 11 and up
Exercise 11 – Circle and line through the origin
Exercise 12 – Tangent to a circle
Show that the line \(y = x + 4\) is tangent to the circle \(x^{2} + y^{2} = 8\) and find the point of tangency.
Exercise 13 – Count the intersections
Use the discriminant of the quadratic you get after substitution to find the number of intersection points of each system.
Exercise 14 – Find the parameter
Exercise 15 – Perimeter and area
Exercise 16 – Two numbers
Exercise 17 – Ball and ramp
A ball’s path is \(y = -x^{2} + 6x\) and a ramp is the line \(y = x + 4\) (both in feet). Find where the ball meets the ramp.
Exercise 18 – Break-even
A school’s revenue is \(R = -x^{2} + 14x\) and its cost is \(C = 2x + 27\), both in hundreds of dollars for \(x\) hundred items. Find the break-even quantities.
Exercise 19 – Find the error
Exercise 20 – Write a system
Show that the system \(y = x^{2} – 4x + 3\) and \(y = x – 1\) has the solutions \((1, 0)\) and \((4, 3)\), and decide which panel above matches it.
Key facts behind this nonlinear systems worksheet
- Substitute the line into the quadratic to get one equation in x.
- Solve the resulting quadratic; then find y for each x.
- D > 0: two intersection points; D = 0: one (tangent); D < 0: none.
- A solution is an ordered pair (x, y).
- The method also works for a line and a circle.
How to use this nonlinear systems worksheet
Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.
Download the nonlinear systems worksheet as a PDF with the button below and keep it with your notes.
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