Systems of linear and quadratic equations

Learning Objectives Identify the possible number of solutions for a linear-quadratic system (zero, one, or two). Solve systems of linear and quadratic equations algebraically using the substitution method. Solve systems of linear and quadratic equations algebraically using the elimination method. Interpret the intersection points of a line and a parabola on a graph as the solutions to the system. Use the discriminant to determine the number of real solutions for a linear-quadratic system. Model and solve a real-world problem using a system of linear and quadratic equations. Verify the solution(s) to a system by substituting them back into the original equations. Imagine a rocket launching in a parabolic arc while a laser beam shoots in a straight line. 🚀 Will their paths...

TermDefinitionExample System of Linear and Quadratic EquationsA set of two equations, one linear and one quadratic, that share the same two variables (usually x and y).y = 2x + 1 (linear) and y = x² - 2x + 4 (quadratic). Solution to a SystemAn ordered pair (x, y) that makes both the linear and the quadratic equations true. Graphically, this is the point where the line and the parabola intersect.For the system y = x + 2 and y = x², the point (2, 4) is a solution because 4 = 2 + 2 and 4 = 2² are both true. Substitution MethodAn algebraic technique where you solve one equation for a variable (e.g., solve for y) and then substitute that expression into the other equation.Given y = x + 1 and y = x² + 3x, you can substitute 'x + 1' for 'y' in the second equation to get x + 1...

General Form of the System Linear: y = mx + b Quadratic: y = ax² + bx + c This is the most common setup. The goal is to find the (x, y) pairs that satisfy both equations simultaneously. The Discriminant for Number of Solutions D = B² - 4AC After using substitution, you will get a new quadratic equation in the form Ax² + Bx + C = 0. The discriminant of THIS new equation tells you the number of solutions: D > 0 means two solutions, D = 0 means one solution, and D < 0 means no real solutions. The Quadratic Formula x = \frac{-B \pm \sqrt{B^2-4AC}}{2A} Use this formula to solve for the x-values of the intersection points from the combined equation Ax² + Bx + C = 0 when it cannot be easily factored.