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Algebra II – Quadratic Systems

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    Study Guide Algebra II – Quadratic Systems 1. Systems of Equations Solved Algebraically What Is a System of Equations? A system of equations is a set of two or more equations that contain the same variables . The goal is to find values of the variables that satisfy all equations at the same time . For example: A solution to a system is usually written as an ordered pair ((x,y)). Methods for Solving Systems Algebraically There are two main methods: 1. Substitution Method 2. Elimination Method Both methods aim to remove one variable so the system becomes easier to solve. Example 1 — Elimination Method Solve the system: Step 1: Multiply the second equation by 2

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    Study Guide Step 2: Add the equations Add them: Step 3: Substitute into the first equation Using Case 1: (x = 2) Solutions:

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    Study Guide Case 2: (x = - 2) Solutions: Final Solutions The system has four solutions : These represent the points where the graphs intersect . Example 2 — Substitution Method Solve:

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    Study Guide Step 1: Solve the second equation for x Step 2: Substitute into the first equation Replace (x): Step 3: Expand Step 4: Move everything to one side Factor: Step 5: Solve for y

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    Study Guide or Step 6: Find x Using (x = y + 2) If (y = 6) If (y = - 8) Final Solutions These are the points where the graphs intersect . Key Ideas to Remember Elimination Method 1. Multiply equations if needed. 2. Add or subtract equations to eliminate one variable. 3. Solve the remaining equation. 4. Substitute back to find the other variable. Substitution Method

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    Study Guide 1. Solve one equation for a variable. 2. Substitute into the other equation. 3. Solve the resulting equation. 4. Substitute back to find the remaining variable. Summary A system of equations involves two or more equations with the same variables. Two common algebraic methods are: Substitution Solve one equation for a variable and substitute into the other. Elimination Add or subtract equations to eliminate a variable. The solutions represent the intersection points of the graphs of the equations. 2. Quiz: Systems of Equations Solved Algebraically 1. Question Solve the system: Answer Choices • (x = - 2, y = 2) • (x = 2, y = - 2) • (x = 3, y = - 2)

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    Study Guide Correct Answer Why This Is Correct From the second equation: Substitute into the first equation: Solving gives: Substitute back: 2. Question Solve the system: Answer Choices • (x = ±3, y = ±4) • (x = ±4, y = ±3) • (x = ±3, y = ±2) Correct Answer

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    Study Guide Why This Is Correct Let: Then: Solve the system: Thus: 3. Question Solve the system: Answer Choices • (a = ±2, b = ±1) • (a = 2, b = ±3) • (a = ±1, b = ±2) Correct Answer Why This Is Correct

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    Study Guide Let: Then: Solving gives: So: 4. Question Solve the system: Answer Choices • (p = ±3, q = ±4) • (p = ±2, q = ±5) • (p = ±5, q = ±2) Correct Answer Why This Is Correct Rewrite equations:

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