In Algebra, and other fields of mathematics, a **variable** is a placeholder, which represents an unknown quantity. Variables are usually represented with a letter. At a simple level, you can think of them like containers that hold numbers we’re trying to figure out.

Variables are powerful, essential tools in the math, science, and computing fields. But before we get into some potential uses of variables, let’s examine some more basic concepts.

Now consider this equation, where the value of 4 is replaced with the variable 𝑥.

Consider the following equation: \(6+4=10\)

\(6+x=10\)

What does the variable \(x\) equal in the above equation?

𝑥 must equal 4, because 4 is the only potential value that satisfies \(6+x=10\)

Notice that the variable 𝑥 is displayed in a lower case stylized script or cursive font. This is commonly how variables are shown in academic materials. You do not need to write them this way, if you don’t know cursive or if your handwriting isn’t the best. A lower case letter written however you normally would write it will suffice, unless your teacher specifically asks you to write them stylized.

Now, look at this equation: \(10-y=5\)

What must the variable 𝑦 equal in the above equation?

It may be helpful to ask yourself the question, 10 minus *what* equals five? Ten minus five equals five.

So the solution to the equation is: \(y=5\)

## Substitution

**Substitution** means replacing one thing with another. The following examples show how to substitute values in for variables.

Given \(x=1\) solve the following expression.

Expression: \(5+x\)

Substitute \(1\) in for \(x\)

\(5+1\)

Then solve.

\(5+1=6\)

\(5+x=6\)

Given \(y=5\) solve the following expression.

Expression: \(y+8\)

Substitute 5 in for \(y\)

\(5+8\)

Solve it.

\(5+8=13\)

\(y+8=13\)

Given \(x=5\) and \(y=2\), solve the following expression:

Expression: \(x\times y\)

Substitute 5 in for \(x\) and 2 in for \(y\), rewrite the expression.

\(5 \times 2\)

Solve it.

\(5 \times 2 = 10\)

\(x \times y = 10\)

**Given**: \(x=10, y=5, z=5\) Solve the following expression

**Expression:** \(5x−3yz\)

**Substitute**: \(5(10)−3(5)5\)

**Solve:**

\(5\times10−3\times5\times5\)

\(50−75\)

\(-25\)

### Review Questions: Substitution

Given \(a=5, b=2, c=7\) evaluate the following expressions.

**r1.** Evaluate

\(5\times a\)

**r2.** Evaluate

\(3\times c-a\)

**r3.** Evaluate

\(b+a+c-5\)