Questions and Answers
What does it mean for a system to be consistent?
To cross somewhere
What is Step 1 of solving a quadratic-linear system?
Solve the linear equation for y.
How do you substitute y in the quadratic equation?
x + 1 = -x^2 + 6x - 3
What is the standard form of the quadratic equation?
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What do you do in Step 4 when solving for x?
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What values of x are found after solving the equation?
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What is the next step after finding x?
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What are the coordinates of the solutions to the quadratic-linear system?
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Study Notes
Quadratic-Linear Systems Overview
- A consistent system is one that has at least one solution, meaning the graphs of the equations intersect at some point.
Solving Quadratic-Linear Systems Algebraically
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Step 1: Solve the Linear Equation for y
- Example: From (y - x = 1), rearranging gives (y = x + 1).
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Step 2: Substitute y into the Quadratic Equation
- Replace (y) in the quadratic (y = -x^2 + 6x - 3) with the linear equation’s expression, resulting in: (x + 1 = -x^2 + 6x - 3).
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Step 3: Write the Quadratic in Standard Form
- Rearranging leads to: (x^2 - 5x + 4 = 0).
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Step 4: Solve for x
- Factoring the quadratic gives ((x - 4)(x - 1) = 0), leading to:
- (x = 4)
- (x = 1)
- Factoring the quadratic gives ((x - 4)(x - 1) = 0), leading to:
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Step 5: Plug x into Either Equation to Find y
- Using the respective (x) values:
- For (x = 4): (y - 4 = 1) yields (y = 5).
- For (x = 1): (y - 1 = 1) gives (y = 2).
- Using the respective (x) values:
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Step 6: Write as Coordinates
- The solutions represented as coordinates are: ((4, 5)) and ((1, 2)).
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Description
This quiz explores the process of solving quadratic-linear systems algebraically. It guides you through each step, from rearranging linear equations to finding the intersection points. Test your understanding of these important algebraic concepts.
