Solving Two-Variable Linear Equations
Understanding Variables and Constants
The Building Blocks of Equations
Think of an equation as a sentence. Just like sentences have nouns and verbs, equations have their own special parts that help us describe relationships between numbers. The two most basic parts are variables and constants.
A variable is a symbol, usually a letter like or , that stands in for a number we don't know yet. It's a placeholder for a value that can change or that we need to figure out. Imagine you have a box, but you don't know how many apples are inside. You could use the letter to represent the unknown number of apples.
Variable
noun
A symbol, typically a letter, used to represent an unknown or changing quantity in a mathematical expression.
Variables are the mysteries of algebra. Our goal is often to find out what number they're hiding.
The Unchanging Numbers
If variables are the unknowns, constants are the parts we do know. A constant is a number whose value is fixed and doesn't change within a problem. In the sentence "I have 2 hands and some number of fingers," the number 2 is a constant. It's a fact. The number of fingers could be a variable if we didn't know it, but the number of hands is fixed.
Constant
noun
A value in a mathematical expression that does not change.
So, if you see an equation like , the numbers 3 and -8 are constants. Their values are set. The letters and are variables because their values can change depending on the situation.
Putting It All Together
Variables and constants work together to form mathematical expressions and equations. Let's look at a common scenario. Imagine you buy a cup of coffee for $3 and a few donuts that each cost $2. The total cost depends on how many donuts you buy.
We can represent this situation with an expression. Let's use the variable to stand for the number of donuts. The cost of the donuts would be . Since the coffee is always $3, the total cost would be:
In this expression, is the variable, while 2 and 3 are constants. The number 2, which is multiplied by the variable, has a special name: a coefficient. It tells us how many of the variable we have.
| Part | Example | Role |
|---|---|---|
| Variable | Represents the unknown number of donuts. | |
| Coefficient | The constant multiplied by the variable. | |
| Constant | A fixed value, the cost of the coffee. |
Soon, you'll be working with problems that have two different variables, known as a system of linear equations with two variables.
Linear equation with two variables
This just means you'll have two mysteries to solve at once, using two variables like and . But the basic building blocks remain the same: variables represent the unknowns, and constants provide the fixed numbers you work with.
In the equation , what does the letter 'x' represent?
A constant is a value in an equation that is always known and does not change.