Real-world examples of linear relationships and equations

📚 Quick Study Guide

• 📏 Definition: A linear relationship is one that can be represented graphically by a straight line.
• ✏️ Equation Form: The most common form is slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• 📈 Slope: The slope ($m$) represents the rate of change; how much $y$ changes for every unit change in $x$. It is calculated as: $m = \frac{\text{change in }y}{\text{change in }x}$.
• 📍 Y-intercept: The y-intercept ($b$) is the value of $y$ when $x = 0$. It's the point where the line crosses the y-axis.
• 💡 Real-World Examples: Think about distance vs. time at a constant speed, or the cost of items at a fixed price per item.
• 📝 Graphing: Plot at least two points that satisfy the equation and draw a straight line through them.

🧪 Practice Quiz

1. A taxi charges an initial fee of $2.50 plus $0.30 per mile. Which equation represents the total cost (y) for x miles?
   1. A. $y = 0.30x + 2.50$
   2. B. $y = 2.50x + 0.30$
   3. C. $y = 2.80x$
   4. D. $y = 2.20x$
2. A phone company charges a monthly fee of $15 plus $0.05 per minute of talk time. If your bill was $22, how many minutes did you talk?
   1. A. 70 minutes
   2. B. 140 minutes
   3. C. 150 minutes
   4. D. 290 minutes
3. Sarah saves $10 per week. She started with $50. Which equation represents her total savings (y) after x weeks?
   1. A. $y = 50x + 10$
   2. B. $y = 10x + 50$
   3. C. $y = 60x$
   4. D. $y = 40x$
4. A rental car costs $25 per day plus a one-time fee of $75. If you rent the car for 5 days, what is the total cost?
   1. A. $100
   2. B. $125
   3. C. $200
   4. D. $250
5. A store sells apples for $2 per pound. Which graph shows the relationship between the weight of the apples (x) and the total cost (y)?
   1. A. A straight line through the origin with a slope of 2.
   2. B. A horizontal line at y = 2.
   3. C. A vertical line at x = 2.
   4. D. A straight line through the origin with a slope of 0.5.
6. A plant grows 1.5 inches per week. If it was initially 4 inches tall, what will be its height after 8 weeks?
   1. A. 8 inches
   2. B. 12 inches
   3. C. 16 inches
   4. D. 20 inches
7. A hot dog vendor sells hot dogs for $3 each. If his daily operating cost is $50, how many hot dogs must he sell to break even?
   1. A. 15
   2. B. 16
   3. C. 17
   4. D. 18