Understanding Variables

In algebra, a variable is like a mystery box.

It’s a symbol, often a letter like x, y, or z, that represents a number we don’t know yet. Variables are incredibly powerful because they allow us to create and solve equations, understand patterns, and describe relationships between numbers. Let’s take a closer look at what variables are and how they work.

Features of a Variable

  • Letters & Symbols

    Variables are often represented by letters (e.g., x, y, z) to stand in for unknown values or quantities.

  • Flexible

    A variable can take on different values depending on the expression or equation.

  • Operations

    Variables represent an unknown number, so you can treat them just like any other number. They can be multiplied, divided, added, and subtracted!

Variables in Action

What are some examples of how we might use a variable?

  • Example #1: Suppose you have a box with some marbles in it, but you don’t know how many. You can call the number of marbles in the box x. Now, instead of saying, “The number of marbles in the box,” you can just use x to represent that number.

  • Example #2: Imagine you’re planning a trip, and you’re unsure how many miles you’ll drive. You can use d to represent the total miles. If your car uses 1 gallon of gas for every 30 miles, the equation d = 30g can tell you how far you can drive with g gallons of gas.

  • Example #3: Think about saving money. If you save $10 each week, but don’t know how many weeks you’ll save, you can use w for the number of weeks. The total savings can be written as 10w, where w represents the time you spend saving.

Using Variables in Simple Equations

Let’s start with a basic example:

You know you have 5 apples, and you buy some more, but you’re not sure how many. Now you have 5 + x apples. Here, x is the variable representing the unknown number of apples you bought.

If someone tells you that you now have 10 apples in total, you can write an equation:

5 + x = 10

To find out how many apples you bought, solve for x:

x = 10 - 5

x = 5

In order to solve for x, you need to isolate the variable. This means moving everything else onto the other side of the equal sign. In this example, we need to subtract 5, which leaves us with x=5. 

Variables in Real-World Problems

Variables are everywhere in real life.

Example 1: Shopping

You’re buying bananas at $0.50 each. If you buy x number of bananas, the total cost is:

0.50x

If you spend $5, you can solve for x:

0.50x = 5

x = 5 ÷ 0.50

x = 10 bananas.

Example 2: Traveling

If you’re driving at 60 miles per hour for t hours, the total distance you’ll travel is:

Distance = 60t

If you drive for 3 hours, substitute t = 3 into the equation:

Distance = 60 × 3 = 180 miles.

Variables in Multi-Step Equations

Sometimes it takes multiple steps to get a variable on its own:

You know that the total cost of a pizza party is $50. Each pizza costs $8, and you also paid a delivery fee of $10. How many pizzas did you order?

Start with an equation:

8p + 10 = 50

Subtract the delivery fee of 10 from both sides:

8p = 40

Divide by 8 to solve for p:

p = 40 ÷ 8 = 5

You ordered 5 pizzas.

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