#Linear Equations and Graphs: Your Ultimate Study Guide 🚀

Hey there! Let's get you prepped for the AP SAT (Digital) with a super focused review of linear equations and graphs. This is a BIG topic, so let’s make sure you’re totally comfortable with it. Think of this as your go-to resource for acing those questions!

#Understanding the Basics

Linear equations and functions are all about straight lines. They show how two variables relate, and mastering them is crucial for the exam. Let's break it down:

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• Slope (m): The rate of change. How much does 'y' change for every change in 'x'?

  • Rise over run: Vertical change / Horizontal change. Think of it as climbing a hill (or going down one!).
  • Positive slope: Line goes up from left to right. ↗️
  • Negative slope: Line goes down from left to right. ↘️
  • Zero slope: Horizontal line. ↔️
  • Undefined slope: Vertical line. ↕️

• Y-intercept (b): Where the line crosses the y-axis. It's the 'y' value when 'x' is zero. Often your starting point or initial value.

#Equations of Lines: Your Toolkit

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• Equation: $$y = mx + b$$
  • m is the slope.
  • b is the y-intercept.
  • Super useful when you know the slope and y-intercept directly from the graph.

#Point-Slope Form

• Equation: $$y - y₁ = m(x - x₁)$$
  • (x₁, y₁) is a point on the line.
  • m is the slope.
  • Great when you have a point and the slope, but not the y-intercept.

#Standard Form

• Equation: $$Ax + By = C$$
  • A, B, and C are constants.
  • Useful for finding intercepts and converting between forms.

#Graphing and Analyzing Lines

#Graphical Representation

• Linear equations always form straight lines.
• The equation dictates the line’s slope and y-intercept.
• X-intercept: Where the line crosses the x-axis (set y = 0).
• Y-intercept: Where the line crosses the y-axis (set x = 0).

#Parallel and Perpendicular Lines

• Parallel lines: Same slope, different y-intercepts. 👯
• Perpendicular lines: Slopes are negative reciprocals (e.g., if one slope is 2, the other is -1/2). ⟂

#Real-World Applications

#Modeling Real-World Situations

• Linear functions model constant change (e.g., cost per item, speed).
• Slope: Rate of change in context (e.g., price increase per year).
• Y-intercept: Initial value or starting point (e.g., initial cost).
• Use equations to predict values, analyze trends, and solve problems.

#Problem-Solving Strategies

• Use equations to find specific values.
• Use graphs to estimate key points (like break-even points).
• Find intersections of lines to see where two scenarios are equal.
• Remember to consider domain and range (e.g., time can’t be negative).

#Final Exam Focus 🎯

Okay, you’re almost there! Here’s what to focus on:

• Master the different forms of linear equations and know when to use each one.
• Understand slope and y-intercept inside and out. They’re the keys to everything!
• Practice interpreting graphs and real-world situations.
• Be ready to identify parallel and perpendicular lines quickly.

#Last-Minute Tips

• Time Management: Don’t get bogged down on one question. If you're stuck, move on and come back.
• Common Pitfalls: Watch out for negative signs and don’t mix up slope and y-intercept.
• Strategies: Draw diagrams, use process of elimination, and always double-check your work.

#Practice Questions

Practice Question

#Multiple Choice Questions

1. A line has a slope of -3 and passes through the point (2, 5). Which of the following is the equation of the line in slope-intercept form? (A) y = -3x + 11 (B) y = -3x - 1 (C) y = 3x + 11 (D) y = -3x + 5

2. Which of the following equations represents a line that is parallel to the line y = 2x - 5 and passes through the point (1, 4)? (A) y = 2x + 2 (B) y = -1/2x + 4.5 (C) y = 2x + 6 (D) y = -1/2x + 2

3. The cost of renting a car is <math-inline>25 per day plus a </math-inline>50 initial fee. Which linear equation represents the total cost (y) for renting the car for (x) days? (A) y = 25x + 50 (B) y = 50x + 25 (C) y = 75x (D) y = 25x - 50

#Free Response Question

A small business owner is analyzing their costs and revenue. The cost (C) to produce x units of a product is given by the equation C = 5x + 100. The revenue (R) from selling x units is given by the equation R = 15x.

(a) Graph both the cost and revenue equations on the same coordinate plane. (2 points)

(b) Determine the break-even point (the point where cost equals revenue) by finding the intersection of the two lines. Show your work. (2 points)

(c) Explain what the slope of the cost equation represents in the context of the problem. (1 point)

(d) If the business owner wants to make a profit, should they produce more or less than the break-even point? Explain your reasoning. (2 points)

Scoring Breakdown:

(a) Graphing:

• 1 point for correctly graphing the cost equation.
• 1 point for correctly graphing the revenue equation.

(b) Break-Even Point:

• 1 point for setting the cost and revenue equations equal to each other (5x + 100 = 15x).
• 1 point for correctly solving for x (x = 10) and finding the corresponding y-value (y=150), stating the break-even point is (10, 150).

(c) Slope Interpretation:

• 1 point for explaining that the slope of the cost equation (5) represents the cost to produce one additional unit.

(d) Profit Analysis:

• 1 point for stating that the business owner should produce more than the break-even point to make a profit.
• 1 point for explaining that beyond the break-even point, the revenue will be greater than the cost.

Alright, you’ve got this! Go into that exam confident and ready to show off your skills. You're well-prepared and totally capable. Good luck! 🍀