Proportional Relationships: Writing Equations

Proportional relationships have a special constant ratio that connects variables!

WHAT IS A PROPORTIONAL RELATIONSHIP? Two quantities x and y are proportional if: - y/x = k (constant ratio) - The relationship can be written as: y = kx - The graph passes through the origin (0, 0)

THE CONSTANT OF PROPORTIONALITY (k): k represents the unit rate or constant ratio

To find k from a table: k = y/x (using any pair of values)

FINDING k FROM A TABLE:

Example Table: | x | y | |---|---| | 2 | 6 | | 4 | 12 | | 6 | 18 |

Find k: k = y/x = 6/2 = 3 (or 12/4 = 3, or 18/6 = 3) Notice k is the SAME for all pairs!

Equation: y = 3x

STEP-BY-STEP PROCESS:

Step 1: Verify it's proportional - Check if y/x is constant for all pairs - OR check if the graph would pass through (0, 0)

Step 2: Calculate k - Choose any (x, y) pair from the table - Divide: k = y/x

Step 3: Write the equation - Use the form: y = kx - Substitute your k value

REAL-WORLD EXAMPLES:

Example 1: Cost of Apples | Pounds (x) | Cost (y) | |-----------|----------| | 2 | $5 | | 4 | $10 | | 6 | $15 |

k = 5/2 = 2.5 (cost per pound) Equation: y = 2.5x or C = 2.5p Meaning: Apples cost $2.50 per pound

Example 2: Distance and Time | Hours (x) | Miles (y) | |----------|----------| | 1 | 60 | | 3 | 180 | | 5 | 300 |

k = 60/1 = 60 (miles per hour) Equation: y = 60x or d = 60t Meaning: Traveling at 60 mph

IDENTIFYING PROPORTIONAL VS NON-PROPORTIONAL:

Proportional: - Constant ratio (y/x) - Passes through origin - Equation: y = kx (no added constant)

NOT Proportional: - Variable ratio - Doesn't pass through origin - Equation: y = kx + b (with b ≠ 0)

INTERPRETING k: The value of k tells you: - Unit rate (price per item, miles per hour) - Slope of the line - How much y changes for each unit of x

APPLICATIONS: - Unit pricing (cost per pound, price per gallon) - Speed (miles per hour, meters per second) - Recipes (cups of flour per batch) - Currency exchange rates - Wages (dollars per hour)

Master writing equations from proportional relationships!

IDENTIFY THE CONSTANT (k): For questions 1-4, find the constant of proportionality k.

1. | x | y |
   |---|---|
   | 3 | 12 |
   | 5 | 20 |
   | 7 | 28 |

2. | x | y |
   |---|---|
   | 4 | 10 |
   | 8 | 20 |
   | 12 | 30 |

3. | x | y |
   |---|---|
   | 2 | 7 |
   | 6 | 21 |
   | 10 | 35 |

4. | x | y |
   |---|---|
   | 5 | 2 |
   | 10 | 4 |
   | 15 | 6 |

WRITE THE EQUATION: For questions 5-8, find k and write the equation y = kx.

5. | x | y |
   |---|---|
   | 1 | 8 |
   | 3 | 24 |
   | 5 | 40 |

6. | x | y |
   |---|---|
   | 2 | 3 |
   | 8 | 12 |
   | 10 | 15 |

7. | Minutes (x) | Words (y) |
   |------------|-----------|
   | 2 | 80 |
   | 5 | 200 |
   | 10 | 400 |

8. | Gallons (x) | Miles (y) |
   |-----------|-----------|
   | 2 | 50 |
   | 5 | 125 |
   | 8 | 200 |

IS IT PROPORTIONAL?
9. Determine if this relationship is proportional:
   | x | y |
   |---|---|
   | 1 | 5 |
   | 2 | 10 |
   | 3 | 16 |

10. Determine if this relationship is proportional:
    | x | y |
    |---|---|
    | 0 | 0 |
    | 4 | 12 |
    | 8 | 24 |

WORD PROBLEMS: 11. A car travels at a constant speed. After 3 hours, it has traveled 195 miles. After 5 hours, 325 miles. Write an equation for distance (d) in terms of time (t).

12. Bananas cost $3.75 for 5 pounds and $7.50 for 10 pounds. Write an equation for cost (C) in terms of pounds (p).

13. A printer prints 120 pages in 4 minutes and 300 pages in 10 minutes. Write an equation for pages (P) in terms of minutes (m).

CHALLENGE: 14. If y = kx and the point (6, 15) is on the line, what is the value of y when x = 10?

15. Create your own proportional relationship table with at least 3 pairs of values, find k, and write the equation.