y=kx & Constant of Proportionality

SECTION 1

Historical Context & Motivation

People have been comparing quantities for thousands of years. Ancient builders needed to know how much stone to order for a wall twice as long. Merchants needed to figure out the cost of five bags of grain if they knew the price of one. The idea of proportional relationships — where two quantities grow at the same rate — is one of the oldest ideas in math.

~1800 BCE

Babylonian Ratio Tables

Ancient Babylonians carved clay tablets listing prices. If 1 basket of barley cost 2 silver coins, they wrote tables showing 2 baskets cost 4, 3 baskets cost 6, and so on.

~300 BCE

Euclid's Proportions

The Greek mathematician Euclid wrote rules about proportions in his famous book Elements. He showed that proportional reasoning applies to geometry, not just numbers.

1600s

Variables Enter Math

Mathematicians like René Descartes began using letters like x and y to stand for unknown numbers. This made it possible to write a single equation for an entire relationship.

Today

y = kx Everywhere

Scientists, engineers, and everyday people use y = kx to model unit prices, speed, recipes, currency exchange, and much more.

Here's the big question this lesson answers: when two quantities always change at the same rate, how can we describe that relationship with one short equation? The answer is y = kx, and the letter k tells us exactly how the two quantities are connected.

SECTION 2

Core Principles & Definitions

Before we dive into the equation, let's lock in some key ideas. These four principles are the building blocks for everything that follows.

Proportional Relationship

Two quantities are proportional when they always have the same ratio. If you double one, the other doubles too.

Constant of Proportionality (k)

The constant of proportionality, written as k, is the number you multiply x by to get y. It stays the same for every pair of values.

The Equation y = kx

This equation says: "y equals k times x." It's the shorthand for any proportional relationship. The k tells you the rate.

Finding k from a Table

To find k, pick any row of a table and divide: k = y ÷ x. If the answer is the same for every row, the relationship is proportional.

✦ KEY TAKEAWAY

Think of k like a recipe multiplier. If a smoothie recipe uses 2 bananas per serving, then k = 2. For 3 servings you need 2 × 3 = 6 bananas. No matter how many servings you make, the rate (2 bananas per serving) never changes. That steady rate is the constant of proportionality.

SECTION 3

Seeing Proportional Relationships

A proportional relationship has a special look on a graph. It always forms a straight line that passes through the origin (0, 0). The steeper the line, the larger the value of k.

Both lines start at the origin (0, 0). The cyan line (k = 2) is steeper because y grows faster. The violet dashed line (k = 1.5) rises more gently. A bigger k always means a steeper line.

Notice that both lines pass through (0, 0). This makes sense: if you buy 0 servings, you need 0 bananas! Every proportional relationship starts at the origin. If a line doesn't go through (0, 0), the relationship is not proportional.

SECTION 4

The Mathematical Framework

Let's look at the equation and the formulas you'll use most often. Don't worry — there are only two main formulas, and they're really the same idea flipped around.

PROPORTIONAL RELATIONSHIP

y = k × x

y = the output (what you're finding), k = the constant of proportionality (the rate), x = the input (what you already know).

FINDING k

k = y ÷ x

Divide any y-value by its matching x-value. If the answer is the same for every pair, then k is that number and the relationship is proportional.

EQUIVALENT RATIO TEST

y₁ ÷ x₁ = y₂ ÷ x₂ = y₃ ÷ x₃ = … = k

Every pair in a proportional table gives the same quotient. That shared quotient is k.

💡 Remember!

The constant of proportionality k is also called the unit rate. If bananas cost $0.50 each, then k = 0.50 and the equation is y = 0.50x, where x is the number of bananas and y is the total cost.

SECTION 5

Finding k in Tables

One of the most common ways you'll see proportional relationships is in a table. Let's look at two tables side by side — one that is proportional and one that is not.

On the left, every row gives y ÷ x = 8, so k = 8 and the equation is y = 8x. On the right, the ratios are all different, so that table does not show a proportional relationship.

The left table might represent a job that pays $8 per hour with no flat starting bonus. The right table might represent a job that pays an hourly rate plus a sign-up bonus. That extra bonus breaks the proportional pattern.

SECTION 6

Worked Example

Let's walk through a full problem from start to finish. Follow each step carefully.

Lemonade Stand Earnings

Step 1 — Read the ProblemMaya sells cups of lemonade. She earns $3 for every cup she sells. Write an equation in the form y = kx. Then find how much she earns after selling 15 cups.

Step 2 — Identify the VariablesLet x = the number of cups sold. Let y = the total earnings in dollars. The rate (dollars per cup) is the constant of proportionality.

x = cups, y = dollars, k = dollars per cup

Step 3 — Find kMaya earns $3 per cup. That means for every 1 cup, y goes up by 3. So k = 3.

k = 3

Step 4 — Write the EquationSubstitute k = 3 into y = kx.

y = 3x

Step 5 — Solve for 15 CupsPlug in x = 15: y = 3 × 15 = 45.

y = $45

Step 6 — Check Your AnswerDoes 45 ÷ 15 equal our k? Yes — 45 ÷ 15 = 3. ✓ The answer makes sense.

Maya earns $45 after selling 15 cups of lemonade.

SECTION 7

Proportional vs. Non-Proportional

Not every relationship between two numbers is proportional. Here's how to tell the difference quickly.

Comparing proportional and non-proportional relationships

Feature	Proportional (y = kx)	Non-Proportional
Graph	Straight line through (0, 0)	Curved line, or straight line that misses (0, 0)
Table	y ÷ x gives the same number every time	y ÷ x gives different numbers
Equation form	y = kx (no added or subtracted number)	y = kx + b, or another form with extra terms
Example	Cost = $5 × (number of tickets)	Cost = $5 × (number of tickets) + $2 fee

✦ KEY TAKEAWAY

Imagine filling identical water bottles from a faucet. If every bottle takes exactly 10 seconds, that's proportional — k = 10 seconds per bottle. But if you also need 5 seconds to turn on the faucet at the start, those first 5 seconds break the proportion. The "+5" is like the extra number in y = kx + b. Proportional means no extra number — just multiply.

SECTION 8

Connection to Linear Equations

In later math courses you'll meet the equation y = mx + b. This is the general equation for any straight line. The equation y = kx is actually a special case of y = mx + b where b = 0. Knowing y = kx inside and out gives you a head start.

y = kx is a building block for y = mx + b

	y = kx (This Lesson)	y = mx + b (Coming Soon)
What it describes	Proportional relationships only	Any straight-line relationship
Slope	k is the slope	m is the slope
y-intercept	Always 0 (line goes through the origin)	b (can be any number)
Example	y = 4x	y = 4x + 3

When you get to slope-intercept form, remember: you already know what the slope means! It's your old friend k, the constant of proportionality. You'll just add one more piece — the y-intercept b — to handle lines that don't start at (0, 0).

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. Read each answer only after you've given it your best shot!

PROBLEM 1 — CONCEPTUAL

In the equation y = kx, what does the letter k represent? Explain in your own words why it's called a "constant."

PROBLEM 2 — BASIC CALCULATION

A car travels at a constant speed. In 2 hours it covers 120 miles. Find the constant of proportionality k (in miles per hour), then write the equation y = kx.

PROBLEM 3 — INTERMEDIATE

A table shows these pairs: (4, 10), (8, 20), (10, 25), (14, 35). Determine whether the relationship is proportional. If it is, state k and write the equation.

PROBLEM 4 — APPLIED

Sophia is baking cookies. Her recipe says she needs 3 cups of flour for every 24 cookies. She wants to make 60 cookies. Use y = kx to find how many cups of flour she needs. (Hint: let x = number of cookies and y = cups of flour.)

PROBLEM 5 — CRITICAL THINKING

Two students each claim they have a proportional relationship. Student A says: "y = 7x." Student B says: "y = 7x + 2." Who is correct? Explain why, and describe what Student B's graph would look like compared to Student A's.

SUMMARY

Lesson Summary

A proportional relationship is one where two quantities always have the same ratio. You can write every proportional relationship as y = kx, where k is the constant of proportionality. To find k, divide any y-value by its matching x-value: k = y ÷ x. If every pair in a table gives the same k, the data is proportional.

On a graph, a proportional relationship always forms a straight line through the origin (0, 0). A larger k makes the line steeper. This concept is the foundation for the slope-intercept form y = mx + b that you'll study next — where y = kx is simply the special case with b = 0.