āļø Writing Equations for Proportional Relationships
Turning words into math symbols
šÆ Introduction
Now that you know about k, let's turn these relationships into equations! Equations are like shortcuts - instead of making a huge table, you write one line that works for ANY value.
All proportional relationships can be written as y = kx, where k is the constant of proportionality.
š” Key Concepts
The Magic Formula: y = kx
Every proportional relationship follows this pattern: y equals k times x. That's it! y = kx
š Real Example:
If candy costs $3 each: cost = 3 Ć number of candies. Using letters: c = 3n (where c = cost, n = number)
š y = kx means: output = (constant) Ć input. It's a multiplication machine!
Choosing Good Variable Names
Instead of always using x and y, use letters that make sense! Use the first letter of what you're measuring.
š Real Example:
For carpet cost: c = 1.5f (c for cost, f for square feet). For distance and time: d = 65t (d for distance, t for time)
š Good variable names help you remember what the equation is about.
Recognizing Proportional Equations
An equation is proportional if it's JUST multiplication: y = (some number) Ć x. Nothing added, nothing subtracted, no powers.
š Real Example:
Proportional: y = 4x, d = 1/3t, c = 0.99n. NOT proportional: y = 4 + x, A = sĀ², y = 36/x
š If you see + or - or Ā² in the equation, it's probably NOT proportional.
š Step-by-Step Examples
Example 1: Writing an equation from words
š Problem: Each square foot of carpet costs $1.50. Write an equation relating cost and square feet.
Identify what you're relating
cost (output) and square feet (input)
Choose variable names
Let c = cost in dollars, f = square feet
Identify k (the rate)
k = $1.50 per square foot = 1.5
Write the equation using y = kx format
c = 1.5f
Check with a value
10 sq ft: c = 1.5 Ć 10 = $15 ā
ā Answer: c = 1.5f
Example 2: Writing an equation from a recipe
š Problem: For every 5 cups of grape juice, mix in 2 cups of peach juice. Write an equation with p for peach and g for grape.
Find the relationship as a rate
2 cups peach per 5 cups grape = 2/5 per cup of grape
Identify k
k = 2/5 or 0.4
Write the equation
p = (2/5)g or p = 0.4g
Check: if g = 5, what is p?
p = (2/5) Ć 5 = 2 ā
ā Answer: p = (2/5)g or p = 0.4g
Example 3: Identifying which equations are proportional
š Problem: Which of these represent proportional relationships? a) K = C + 273 b) s = (1/4)p c) V = sĀ³ d) p = 0.4g
Check equation (a): K = C + 273
Has '+ 273' - there's addition! NOT proportional.
Check equation (b): s = (1/4)p
Just multiplication by 1/4. This IS proportional! k = 1/4
Check equation (c): V = sĀ³
sĀ³ means s Ć s Ć s. This is a power, NOT proportional.
Check equation (d): p = 0.4g
Just multiplication by 0.4. This IS proportional! k = 0.4
ā Answer: b and d are proportional
ā ļø Common Mistakes to Avoid
ā Adding extra stuff to the equation
Why it's wrong: y = 3x + 2 is NOT proportional because of the + 2. Proportional equations ONLY have multiplication.
ā How to avoid: Your equation should look like y = (number)x with nothing else. Just one letter, one number, and a multiplication.
ā Confusing which variable is x and which is y
Why it's wrong: If you mix them up, your k will be upside down (you'll get 1/k instead of k).
ā How to avoid: The variable you multiply BY k is x (the input). The variable that EQUALS something is y (the output).
ā Thinking y = x/3 is different from y = (1/3)x
Why it's wrong: They're the same! Dividing by 3 is the same as multiplying by 1/3. Both have k = 1/3.
ā How to avoid: Remember: dividing by a number is the same as multiplying by its reciprocal.
You're turning into an equation-writing machine! š¤ Remember, equations are just a quick way to describe a relationship. Once you write one, you can find ANY value you need!