Solving Linear-Quadratic Systems Assignment

What is an assignment?

What are the solutions to the system of equations: y = x² + x - 2 and y = -x + 1?

For which value of k does the system have no real number solutions?

How many real number solutions does the system y + x = 19 - x² and x + y = 80 have?

What is the second solution for the system y - 10 = 11x + x² and y - 12x = 30?

Which statement is true regarding David's system of equations?

Which statements are true about Jordan's system of equations y = 2x² + 3 and y - x = 6? (Select all that apply)

Which systems of equations have no real number solutions? (Select all that apply)

How can you determine the number of real number solutions without graphing?

Which represents the solution(s) of the system y = -x² + 6x + 16 and y = -4x + 37?

Which represents the solution(s) of the system y = x² - 2x - 3 and y = -x + 3?

If one of the solutions of the system has an x-value of -4, what is its corresponding integer y-value?

Which represents the solution(s) of the system y = x² - 4x - 21 and y = -5x - 22?

Which represents the solution(s) of the system y = x² - 6x + 12 and y = 2x - 4?

Which graph most likely shows a system of equations with no solutions?

Solving Linear-Quadratic Systems

To find solutions for the equations (y = x^2 + x - 2) and (y = -x + 1), the solution set is ((-3, 4)) and ((1, 0)).
A system of equations (y = x^2) and (y = x + k) will have:
- No real solutions when (k = -2),
- One real solution when (k = -0.25),
- Two real solutions when (k = 2).
When solving (y + x = 19 - x^2) and (x + y = 80) algebraically, it results in 0 real solutions.
Solving (y - 10 = 11x + x^2) and (y - 12x = 30) yields solutions ((-4, -18)) and ((5, 90)).
In the system (-4x - 7 = y) and (x^2 - 2x - 6 = y), there is one unique real number solution at ((-1, -3)).
In the system (y = 2x^2 + 3) and (y - x = 6):
- The quadratic equation is in standard form.
- It can be rewritten as (2x^2 - x - 3 = 0).
- This system has two real number solutions.
For systems without real number solutions:
- (y = x^2 + 4x + 7) and (y = 2),
- (y = -x^2 - 3) and (y = 9 + 2x),
- (y = -3x - 6) and (y = 2x^2 - 7x) are examples.
Determining the number of real solutions in a linear-quadratic system can be done using the discriminant:
- If negative, there are no real solutions.
- If zero, there is one real solution.
- If positive, there are two real solutions.
The algebraic solution for the system (y = -x^2 + 6x + 16) and (y = -4x + 37) results in ((3, 25)) and ((7, 9)).
Solving (y = x^2 - 2x - 3) and (y = -x + 3) gives solutions ((3, 0)) and ((-2, 5)).
The integer y-value corresponding to (x = -4) in a given graph is (-3).
The equation (y = x^2 - 4x - 21) and (y = -5x - 22) shows no solutions.
The first steps in solving (y = x^2 - 6x + 12) and (y = 2x - 4) yield the solution ((4, 4)).
A graph of a system of equations with no solutions is typically represented by two parallel lines.

This quiz focuses on solving linear-quadratic systems using a graphing calculator. Test your skills by finding the solutions to specific equations and verifying your answers. Explore various problems to deepen your understanding of the concepts involved.