AP SAT (Digital) Math: Systems of Linear Equations - Your Night-Before Guide

Hey there, future math master! Let's get you prepped and confident for the AP SAT (Digital) Math section on systems of linear equations. This guide is designed to be your quick, go-to resource for a final review. Let's make sure everything clicks!

1. Setting Up Linear Equations from Word Problems

1.1 Identifying and Representing Unknown Quantities

• What are they? Systems of linear equations are just two or more equations with the same variables. Think of them as a puzzle where you need to find the values of those variables.
• Word Problem Clue: If a word problem has two unknown things that are related, that's your cue to use a system of equations!
• The Game Plan:
  • Assign variables (usually x and y) to represent the unknowns. 💡
  • Read the problem super carefully, noting how these unknowns relate to each other or to known numbers.
  • Translate those words into math equations.
• Example: "The sum of two numbers is 10" becomes x + y = 10

1.2 Creating Accurate Equations

• Key Rule: The number of equations needs to match the number of unknowns. Two unknowns? You need two equations!
• Double-Check: Make sure your equations actually match the relationships described in the problem.
• Consistency is Key: Ensure your equations make sense with the problem statement.
• Don't Miss Anything: Include all the relevant info from the problem in your equations.

2. Solving Linear Equations for Word Problems

2.1 Solving Methods

• Your Toolkit:
  • Substitution: Solve one equation for a variable and plug that expression into the other equation.
  • Elimination: Add or subtract equations to get rid of one variable, then solve for the other.
  • Graphing: Plot both equations and find where they intersect. (Less common on the digital SAT but good to know!)

2.2 Interpreting and Verifying Solutions

• Context is King: What do your x and y values actually mean in the problem?
• What do they represent? The solution values are the unknown quantities that make the problem's conditions true.
• Rounding: If your answer is a decimal or fraction, round appropriately based on the problem.
• The Ultimate Test: Plug your answers back into the original equations to make sure they work!
• Does it make sense? Verify that your solution satisfies all the conditions in the problem.
• Special Cases:
  • No solution? (Parallel lines) This means the scenario in the problem is impossible.
  • Infinite solutions? (Coincident lines) The equations are essentially the same, and many combinations work.

2.3 Applying Solutions to Real-World Scenarios

• Practical Meaning: How does your math answer relate to the real-world situation in the problem?
• The Big Picture: Explain how your solution answers the original question.
• Units Matter: Include units in your final answer (e.g., dollars, hours, etc.).
• Realistic Constraints: Consider if your answer makes sense in the real world. Can you have a negative number of apples?
• Assumptions: Be aware of any assumptions you made while solving the problem.

3. Final Exam Focus

• High-Priority Topics:
  • Setting up equations from word problems.
  • Solving systems using substitution and elimination.
  • Interpreting solutions within the context of the problem.
• Common Question Types:
  • Problems involving rates, mixtures, and geometry.
  • Questions that require you to interpret the meaning of the solution.
  • Problems that combine linear systems with other concepts, like linear inequalities.
• Last-Minute Tips:
  • Time Management: Practice setting up equations quickly. Don't spend too long on one problem.
  • Common Pitfalls: Watch out for sign errors and incorrect interpretations of word problems.
  • Strategy: If you're stuck, try to solve for one variable and see if that helps.

4. Practice Questions

Practice Question

Multiple Choice Questions

1. A movie theater sells tickets for $10 for adults and $6 for children. If a total of 250 tickets were sold and the total revenue was $1900, how many adult tickets were sold? (A) 100 (B) 125 (C) 150 (D) 175

2. The sum of two numbers is 30. The larger number is 4 more than twice the smaller number. What is the larger number? (A) 10 (B) 12 (C) 20 (D) 22

Free Response Question

3. A farmer sells apples and oranges at a local market. On one particular day, he sells 15 apples and 10 oranges for a total of $25. The next day, he sells 20 apples and 15 oranges for $34.50. Assume the price of each apple and orange remains the same on both days.

   (a) Write a system of linear equations to represent this situation. (2 points) (b) What is the price of one apple and one orange? (4 points) (c) If the farmer sells 25 apples and 20 oranges on the third day, what is his total revenue? (2 points)

Scoring Breakdown:

(a) System of Linear Equations (2 points)

• 1 point for correctly defining variables (e.g., let 'a' be the price of one apple and 'o' be the price of one orange)
• 1 point for correctly writing the two equations: $$15a + 10o = 25$$ $$20a + 15o = 34.50$$

(b) Price of One Apple and One Orange (4 points)

• 1 point for choosing a correct method (substitution or elimination)
• 1 point for correctly solving for one variable
• 1 point for correctly solving for the second variable
• 1 point for stating the price of one apple and one orange with units (e.g., Apple: $0.80, Orange: $1.30)

(c) Total Revenue on the Third Day (2 points)

• 1 point for setting up the correct calculation: 25 * price of apple + 20 * price of orange
• 1 point for calculating the correct total revenue with units (e.g., $46)

Alright, you've got this! Remember, the key is to break down the problem, translate it into math, and then solve. You're well-prepared, and you're going to do great! Go get 'em!