Linear and quadratic systems
Brian Hall
6 min read
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Study Guide Overview
This study guide covers solving systems of linear equations (substitution, elimination, graphing), analyzing types of solutions (one, none, infinite), and solving linear and quadratic systems (algebraically and graphically). It also explains how to use the discriminant to determine the number of solutions for mixed systems and touches upon real-world applications.
#Linear and Quadratic Systems: Your Ultimate Guide 🚀
Hey! Let's make sure you're totally ready for anything the AP SAT (Digital) throws at you on linear and quadratic systems. This guide is designed to be your go-to resource for a quick, confident review. Let's dive in!
#Solving Systems of Linear Equations
Linear systems are all about finding where lines meet. Think of it as a 'Where's Waldo?' but with equations!
### Methods for Solving Linear Systems
- A system of linear equations is just a set of two or more equations with the same variables. - The solutions? They're the points where the lines cross on a graph. - **Substitution:** Solve one equation for one variable, then plug that into the other equation. - **Elimination:** Add or subtract the equations to get rid of one variable. - **Graphing:** Plot the lines and see where they intersect. - Remember: The intersection point (x, y) gives you the solution.
### Practical Applications
- These systems model real-world stuff, like where supply meets demand. - They're also used in optimization problems to find the best solutions. - Even computer graphics use them to find where lines and planes intersect.
## Solutions to Linear Systems
### Types of Solutions
- **One solution:** Lines intersect at one point. - **No solution:** Lines are parallel and never meet. - **Infinitely many solutions:** Lines are the same, just written differently (they overlap completely).
### Determining Solution Types
- Compare the **slopes** and...
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