Proportional Relationships: Writing Equations
Discover the power of proportional relationships! When two quantities are proportional, they maintain a constant ratio. Learn to find this constant and write equations that model real-world situations.
Concept
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)
Try it
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.