Solve Systems of Linear and Quadratic Equations Algebraically - Algebra-1

1. Fundamental Concepts

System of Linear and Quadratic Equations: A system composed of one linear equation (in the form or ) and one quadratic equation (in the form , ).
Algebraic Solution Meaning: Solving the system algebraically involves finding the x- and y-values that satisfy both equations simultaneously. These values correspond to the coordinates of the intersection points of the line and parabola (geometrically).

Substitution Method: The primary algebraic technique, where one variable is expressed from one equation and substituted into the other. For systems with y-expressions, substitute the linear equation into the quadratic equation to eliminate y and form a single quadratic equation in x.
Quadratic Equation in Standard Form: After substitution, the resulting equation is simplified to . The discriminant ( ) determines the number of solutions:
- : 2 real solutions (two intersection points).
- : 1 real solution (tangent, one intersection point).
- : 0 real solutions (no intersection).
Solving for y: Once x-values are found, substitute them back into the linear equation (simpler to compute) to find corresponding y-values.

System:
Steps:
1. Substitute into the quadratic equation:
2. Rearrange into standard form:
3. Solve the quadratic equation (factor or quadratic formula): or
4. Find y using :
  - For :
  - For :
Solutions: and

System:
Steps:
1. Substitute into the quadratic equation:
2. Rearrange:
3. Solve: or
4. Find y:
  - For :
  - For :
Solutions: and

System:
Steps:
1. Substitute into the quadratic equation:
2. Rearrange:
3. Calculate discriminant:
4. Conclusion: No real solutions (no intersection).

Substitute to Eliminate a Variable: Since both equations often solve for y, substitute the linear equation into the quadratic equation to form a single equation in x.
Simplify to Standard Quadratic Form: Rearrange terms to by moving all terms to one side and combining like terms.
Solve the Quadratic Equation: Use factoring (if possible), completing the square, or the quadratic formula () to find x-values.
Find Corresponding y-Values: Substitute each x-solution back into the linear equation (simpler calculation) to find y.
Check Solutions: Verify all pairs in both original equations to ensure they satisfy both.
Interpret the Discriminant: Use to predict the number of solutions before solving, which helps confirm results.