Linear-Quadratic System Definition
A linear-quadratic system definition describes a system of two equations where one is linear (graphing as a straight line) and one is quadratic (graphing as a parabola), studied in Grade 11 Algebra 1 enVision Chapter 9. The general form pairs y = ax^2 + bx + c with y = mx + d. A line can intersect a parabola at 0, 1, or 2 points, so the system can have zero, one, or two solutions. For example, the system y = x^2 - 4x + 5 and y = x + 1 pairs a parabola with a line. Solutions are the intersection coordinates.
Key Concepts
Property A linear quadratic system of equations consists of one linear equation and one quadratic equation. The general form of such a system is: $$ \begin{cases} y = ax^2 + bx + c \\ y = mx + d \end{cases} $$.
Examples A system with a parabola and a line: $$ \begin{cases} y = x^2 4x + 5 \\ y = x + 1 \end{cases} $$ A system where the linear equation is not in slope intercept form: $$ \begin{cases} y = x^2 + 6 \\ 2x + y = 3 \end{cases} $$.
Explanation A linear quadratic system pairs a quadratic equation, which graphs as a parabola, with a linear equation, which graphs as a straight line. The solutions to the system are the points where the parabola and the line intersect. Because a line can cross a parabola in at most two places, there can be zero, one, or two real solutions. These systems model situations where a linear path or rate interacts with a parabolic trajectory.
Common Questions
What is a linear-quadratic system?
A system of two equations where one is linear (a line) and one is quadratic (a parabola). Solutions are the points where the two graphs intersect.
How many solutions can a linear-quadratic system have?
Zero, one, or two. A line can miss the parabola entirely (0 solutions), be tangent to it (1 solution), or cross it at two points (2 solutions).
Give an example of a linear-quadratic system.
The system y = x^2 - 4x + 5 and y = x + 1, where the first equation is a parabola and the second is a line.
What does it mean if a linear-quadratic system has one solution?
The line is tangent to the parabola — it touches at exactly one point without crossing through it.
Can the linear equation in the system be in a form other than slope-intercept?
Yes. For example, 2x + y = 3 is a valid linear equation in the system, even though it is not in y = mx + b form.
What are the solutions to a linear-quadratic system?
Ordered pairs (x, y) that satisfy both equations simultaneously — the intersection points of the parabola and the line.