A linear-quadratic system of equations is a set of two equations, one linear and one quadratic, that are solved together to find the common solutions.
The general form of a linear-quadratic system is:
- Linear equation: \$y = mx + c\$
- Quadratic equation: \$y = ax^2 + bx + c\$
The solutionto a linear-quadratic systemis the point(s) \$(x, y)\$ where the linearand quadraticequations intersect.
Solving Linear-Quadratic Systems Algebraically
To solve a linear-quadratic system algebraically, follow these steps:
- Set the equations equal to each other: Since both equations are equal to \$y\$, set the right-hand sides equal:
- \$\$mx + c = ax^2 + bx + c\$\$
- Rearrange the equation: Move all terms to one side to form a quadratic equation:
- \$\$ax^2 + (b - m)x + (c - c) = 0\$\$
- Simplify to:
- \$\$ax^2 + (b - m)x = 0\$\$
- Solve the quadratic equation: Use methods like factoring, completing the square, or the quadratic formula to find the \$x\$-values.
- Find the corresponding \$y\$-values: Substitute the \$x\$-values back into either the linear or quadratic equation to find the \$y\$-values.
How does the choice of solving method (algebraic vs. graphical) affect our understanding of the solutions to a system of equations? What are the advantages and limitations of each approach?
