Linear and Quadratic Systems: Foundations

[Video: Solving Linear-Quadratic Systems]

Learn how to solve systems involving linear and quadratic equations, and interpret their solutions graphically.

Understanding Linear-Quadratic Systems

A system consisting of one linear equation and one quadratic equation.

[Graph showing a line intersecting a parabola at two points]

Figure 1: A linear-quadratic system with two solutions

Possible Solution Scenarios

Two solutions: Line intersects parabola twice
One solution: Line is tangent to parabola
No solutions: Line doesn't intersect parabola

Solving Methods

Substitution Method

Solve the linear equation for one variable
Substitute into the quadratic equation
Solve the resulting quadratic equation
Find corresponding y-values
Write solutions as ordered pairs

Example: Solve y = x² - 2x + 1 and y = 2x - 3

1. Substitute: 2x - 3 = x² - 2x + 1

2. Rearrange: x² - 4x + 4 = 0

3. Solve: (x - 2)² = 0 → x = 2

4. Find y: y = 2(2) - 3 = 1

5. Solution: (2, 1) [one solution]

Graphical Interpretation

The solutions represent points of intersection between the line and parabola.

[Graph showing different intersection scenarios]

Figure 2: Different solution cases visualized

Special Cases

No Real Solutions

When the resulting quadratic equation has a negative discriminant (b² - 4ac < 0).

Example: y = x² + 1 and y = x - 2

Substitute: x - 2 = x² + 1 → x² - x + 3 = 0

Discriminant: (-1)² - 4(1)(3) = -11 < 0

No real solutions (graphs don't intersect)

Key Takeaways

Substitution is the most reliable algebraic method
The discriminant predicts number of solutions
Always check solutions in both original equations
Graphical understanding helps verify answers
On SAT, these often appear in word problem contexts

Practice Question

A system of equations is given by y = x² - 4x + 5 and y = x + 1. What are the solutions to this system?

A) (1, 2) and (4, 5)

B) (2, 3) only

C) (1, 2) only

D) (1, 2) and (3, 4)

Follow these steps:

Set the two expressions for y equal to each other
Solve the resulting quadratic equation
Find the corresponding y-values for each x-solution
Verify your solutions in both equations

Step-by-Step Solution:

1. Set equal: x² - 4x + 5 = x + 1

2. Rearrange: x² - 5x + 4 = 0

3. Factor: (x - 1)(x - 4) = 0

4. Solutions: x = 1 and x = 4

5. Find y-values:

For x=1: y = 1 + 1 = 2 → (1, 2)

For x=4: y = 4 + 1 = 5 → (4, 5)

6. Verify both points satisfy both equations

The correct answer is A) (1, 2) and (4, 5).