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Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

Using Variables to Show Relationships Between Quantities

Playing a racing video game. The longer you hold down the gas button, the farther your car goes. Time changes, and distance changes with it. These two quantities are connected. This kind of connection is what we study in algebra with independent and dependent variables.

In this lesson, you learn how to:

• Use variables to represent two related quantities
• Tell which quantity is independent and which is dependent
• Write an equation to show how the two are related
• Organize values in a table
• Graph the relationship on a coordinate plane and connect the graph, table, and equation

The cool part: Once you understand these ideas, you can describe many real-world situations—like earning money, moving at a steady speed, or filling a water bottle—using math.

1. Quantities That Change Together

A quantity is something you can measure or count, like time, distance, money, or number of items.

Sometimes, when one quantity changes, another changes in a predictable way. For example:

• The more hours you work at a job, the more money you earn.
• The more time you spend riding a bike at a steady speed, the farther you travel.
• The more songs you download, the more storage space you use on your device.

These are all situations where two quantities are related.

2. Independent and Dependent Variables

To describe these relationships, we use variables, usually letters like x and y.

• The independent variable is the quantity you choose or control. It is the “input.”
• The dependent variable is the quantity that changes because of the independent variable. It is the “output.”

A helpful way to think about it:

• The dependent variable depends on the independent variable.

Common choices:

• Time is often the independent variable, because other things (distance, amount of water, amount of money) often depend on time.
• Cost is often the dependent variable, because it depends on how many items you buy.

We often use:

• \(x\) for the independent variable (input)
• \(y\) for the dependent variable (output)

But we can also use letters that match the meaning, like \(t\) for time or \(d\) for distance.

3. Example of a Relationship: Constant Speed

Suppose a car travels at a constant speed of 65 miles per hour on a highway.

• The time in hours is the independent variable.
• The distance in miles is the dependent variable.

We can represent the relationship between distance and time with the equation \(d = 65t\). Here:

• \(d\) is the distance (dependent variable).
• \(t\) is the time in hours (independent variable).
• 65 is the speed in miles per hour.

This equation tells us: distance equals 65 times the time. If you know the time, you can find the distance.

The relationship between distance and time for this car is shown in [Figure 1].

4. Using Tables to Show the Relationship

A table is a clear way to list values of the independent and dependent variables. For the car traveling at 65 miles per hour, we can choose some times and calculate distances.

Use the equation \(d = 65t\).

• If \(t = 1\) hour, \(d = 65 \, \textrm{miles}\).
• If \(t = 2\) hours, \(d = 130 \, \textrm{miles}\).
• If \(t = 3\) hours, \(d = 195 \, \textrm{miles}\).
• If \(t = 4\) hours, \(d = 260 \, \textrm{miles}\).

We can organize this in a table:

Table: Time and Distance for \(d = 65t\)

• Time \(t\) (hours): 0, 1, 2, 3, 4
• Distance \(d\) (miles): 0, 65, 130, 195, 260

Notice:

• When \(t = 0\), the distance is 0. If you haven’t driven any time, you haven’t gone anywhere.
• Each time the time increases by 1 hour, the distance increases by 65 miles.

This repeated pattern shows the constant speed. The table of values is represented visually as a set of points on a coordinate grid in [Figure 2].

5. Graphing the Relationship on a Coordinate Plane

A graph lets you see the relationship between two variables as a picture.

• The horizontal axis (x-axis) usually represents the independent variable.
• The vertical axis (y-axis) usually represents the dependent variable.

For the car example:

• Horizontal axis: time \(t\) in hours
• Vertical axis: distance \(d\) in miles

We plot each pair \((t, d)\) as a point. For example, \((2, 130)\) means that after 2 hours, the car has traveled 130 miles.

When we plot all the points from the table, they form a straight line, as shown in [Figure 2]. This straight line shows a constant rate of change: the car travels the same number of miles every hour.

Important ideas from the graph:

• The line goes through the origin \((0, 0)\). That makes sense, because at time 0, the distance is 0.
• The graph is a straight line. That tells us the speed is constant (the same every hour).
• Every point on the line fits the equation \(d = 65t\).

6. Connecting Equation, Table, and Graph

The equation, table, and graph all describe the same relationship in different ways:

• Equation \(d = 65t\): a rule that tells how to get distance from time.
• Table: a list of example pairs that follow the rule.
• Graph: a picture of all the pairs, showing a pattern (a straight line).

If you know any one of these (equation, table, or graph), you can usually find the others. That is one of the powerful things about algebra.

7. More Real-World Relationships

Many everyday situations can be modeled in a similar way.

7.1 Earning Money at a Constant Rate

Suppose you walk your neighbor’s dog and earn 8 dollars per hour.

• The independent variable: time in hours, call it \(h\).
• The dependent variable: total money you earn, call it \(M\).

Every hour you walk the dog, you earn 8 more dollars. The equation is:

\(M = 8h\)

Some values:

• \(h = 1\) → \(M = 8 \, \textrm{dollars}\)
• \(h = 2\) → \(M = 16 \, \textrm{dollars}\)
• \(h = 3\) → \(M = 24 \, \textrm{dollars}\)

If you made a table and a graph, it would look similar in style to the car example, but with different numbers.

7.2 Filling a Water Bottle

Imagine you fill a water bottle from a tap that pours water at a constant rate: 50 milliliters per second.

• Independent variable: time in seconds \(t\).
• Dependent variable: amount of water in milliliters \(w\).

Equation: \(w = 50t\).

Again, time is independent, the amount of water is dependent, and the graph is a straight line going through \((0, 0)\).

7.3 Buying Songs or Apps

Suppose each song you buy from an online store costs 2 dollars. The number of songs affects the total cost.

• Independent variable: number of songs \(s\).
• Dependent variable: total cost \(C\).

Equation: \(C = 2s\).

Here, the graph still shows a straight line, but the horizontal axis is “number of songs” instead of time. The relationship between how many items you buy and the total cost is very common in real life.

8. Solved Example 1: Bike Ride at Constant Speed

Situation: You ride your bike at a constant speed of 12 miles per hour.

Step 1: Choose variables.

• Let \(t\) be the time in hours (independent variable).
• Let \(d\) be the distance in miles (dependent variable).

Step 2: Write an equation.

Each hour, you travel 12 miles. So distance equals 12 times time.

Equation: \(d = 12t\)

Step 3: Make a table of values.

• \(t = 0\) → \(d = 12 \times 0 = 0\)
• \(t = 1\) → \(d = 12 \times 1 = 12\)
• \(t = 2\) → \(d = 12 \times 2 = 24\)
• \(t = 3\) → \(d = 12 \times 3 = 36\)

Step 4: Describe the graph.

• The graph uses time \(t\) on the horizontal axis and distance \(d\) on the vertical axis.
• The points \((0, 0)\), \((1, 12)\), \((2, 24)\), \((3, 36)\) lie on a straight line.
• The line shows that every extra hour increases the distance by 12 miles.

This example has the same structure as the car problem, but a different speed.

9. Solved Example 2: Video Game Points

Situation: In a video game, you get 50 points for every level you complete. After a while, you notice the pattern and want to write an equation.

Step 1: Choose variables.

• Let \(L\) be the number of levels completed (independent variable).
• Let \(P\) be the total number of points (dependent variable).

Step 2: Write an equation.

You get 50 points for each level. So the total points equal 50 times the number of levels.

Equation: \(P = 50L\)

Step 3: Make a table of values.

• \(L = 0\) → \(P = 50 \times 0 = 0\)
• \(L = 1\) → \(P = 50 \times 1 = 50\)
• \(L = 2\) → \(P = 50 \times 2 = 100\)
• \(L = 3\) → \(P = 50 \times 3 = 150\)
• \(L = 4\) → \(P = 50 \times 4 = 200\)

Step 4: Describe the graph.

• Horizontal axis: number of levels \(L\).
• Vertical axis: total points \(P\).
• The points \((0, 0)\), \((1, 50)\), \((2, 100)\), \((3, 150)\), \((4, 200)\) lie on a straight line.
• The line shows a constant rate: every new level adds 50 points.

The relationship between levels and points is visualized as a line of equally spaced points, as shown in [Figure 3].

10. Solved Example 3: Renting a Scooter

Situation: A scooter rental shop charges 5 dollars per hour. You want to model the total cost as a function of time.

Step 1: Choose variables.

• Let \(h\) be the number of hours you rent the scooter (independent variable).
• Let \(C\) be the total cost in dollars (dependent variable).

Step 2: Write an equation.

You pay 5 dollars for each hour. So the total cost equals 5 times the number of hours.

Equation: \(C = 5h\)

Step 3: Make a table of values.

• \(h = 0\) → \(C = 5 \times 0 = 0\) (no hours, no cost)
• \(h = 1\) → \(C = 5 \times 1 = 5\)
• \(h = 2\) → \(C = 5 \times 2 = 10\)
• \(h = 3\) → \(C = 5 \times 3 = 15\)
• \(h = 4\) → \(C = 5 \times 4 = 20\)

Step 4: Analyze the relationship.

• Independent variable: hours \(h\); you can choose how long to rent.
• Dependent variable: cost \(C\); it depends on \(h\).
• For each increase of 1 hour, the cost increases by 5 dollars. This is the constant rate (or unit rate).
• If you graphed the points, you would get a straight line going through the origin, just like the previous examples.

11. How to Tell Which Variable Is Which

When you read a problem, use these questions to identify the independent and dependent variables:

1. Which quantity is chosen or controlled? That is usually the independent variable.
2. Which quantity changes as a result? That is the dependent variable.
3. Can you say “_____ depends on _____”? The first blank is the dependent variable; the second is the independent variable.

Examples:

• “Distance depends on time.” → distance is dependent; time is independent.
• “Total cost depends on number of items.” → total cost is dependent; number of items is independent.
• “Number of followers on a channel depends on days since launch.” → followers are dependent; days are independent.

12. Real-World Applications

These relationships are everywhere in daily life and technology:

• Sports: A runner’s distance depends on how long they run at a certain pace.
• Transportation: Taxi fares or ride-share costs often depend on time and distance.
• Cooking: The amount of ingredients needed can depend on the number of servings.
• Saving money: The amount saved depends on how many weeks you save the same amount.
• Streaming data: The amount of data used can depend on how many minutes you watch videos.

Whenever you see “for every” or “per” (like “miles per hour” or “dollars per song”), you are looking at a rate that can be modeled with an equation like \(y = kx\), where \(k\) is the constant rate.

13. Key Points to Remember

Independent and dependent variables help you describe how two quantities are related. The independent variable is the input you choose; the dependent variable is the output that changes in response. You can use equations, tables, and graphs to represent the same relationship in different but connected ways. For many real-world situations with a constant rate, the relationship can be written in the form \(y = kx\), such as \(d = 65t\) for distance and time at a constant speed. Tables list matching input-output pairs that follow the rule, and graphs show these pairs as points forming a straight line through the origin. Understanding these connections lets you analyze patterns, make predictions, and explain how one quantity changes when another changes.