A mixture of explanations, examples, and practice problems to have you evaluating expressions with one variable in no time!

How to evaluate an expression with one variable

Let's say we want to evaluate the expression a, plus, 4. Well, first we need to know the value of the variable a. For example, to evaluate the expression when start color blueD, a, equals, 1, end color blueD, we just replace start color blueD, a, end color blueD with start color blueD, 1, end color blueD:
empty space, start color blueD, a, end color blueD, plus, 4
equals, start color blueD, 1, end color blueD, plus, 4, space, space, space, space, space, space, space, space, start color gray, R, e, p, l, a, c, e, space, start color blueD, a, end color blueD, space, w, i, t, h, space, start color blueD, 1, end color blueD, point, end color gray
equals, 5
So, the expression a, plus, 4 equals 5 when a, equals, 1.
We can just as easily evaluate a, plus, 4 when start color blueD, a, equals, 5, end color blueD:
empty space, start color blueD, a, end color blueD, plus, 4
equals, start color blueD, 5, end color blueD, plus, 4, space, space, space, space, space, space, space, space, start color gray, R, e, p, l, a, c, e, space, start color blueD, a, end color blueD, space, w, i, t, h, space, start color blueD, 5, end color blueD, point, end color gray
equals, 9
So, the expression a, plus, 4 equals 9 when a, equals, 5.

Evaluating an expression with multiplication

Here's a problem you might see:
Evaluate 3, x when x, equals, 5.
Notice how the number 3 is right next to the variable x in the expression 3, x. This means "3 times x". The reason we do this is because the old way of showing multiplication with the symbol times looks confusingly similar to the variable x.
Okay, so now let's solve the problem:
empty space, 3, start color blueD, x, end color blueD
equals, 3, dot, start color blueD, 5, end color blueD, space, space, space, space, space, space, space, space, R, e, p, l, a, c, e, space, start color blueD, x, end color blueD, space, w, i, t, h, space, start color blueD, 5, end color blueD, point
equals, 15
So, the expression 3, x equals 15 when x, equals, 5.

New ways to show multiplication

Hold on a second! Did you notice that we wrote "3 times start color blueD, 5, end color blueD" as 3, dot, start color blueD, 5, end color blueD instead of as 3, times, start color blueD, 5, end color blueD? Using a dot instead of the symbol times is another new way of showing multiplication:
3, dot, start color blueD, 5, end color blueD, equals, 15
Parentheses can also be used to show multiplication:
3, left parenthesis, start color blueD, 5, end color blueD, right parenthesis, equals, 15
Let's summarize the new ways of showing multiplication that we learned.
Old wayNew way
With a variable3, times, x3, x
Without variable3, times, 53, dot, 5 or 3, left parenthesis, 5, right parenthesis

Evaluating expressions where order of operations matter

For more complex expressions, we'll have to be sure to pay close attention to order of operations. Let's take a look at an example:
Evaluate 5, plus, 3, e when start color blueD, e, equals, 4, end color blueD.
empty space, 5, plus, 3, start color blueD, e, end color blueD
equals, 5, plus, 3, dot, start color blueD, 4, end color blueD, space, space, space, space, space, space, space, space, start color gray, R, e, p, l, a, c, e, space, start color blueD, e, end color blueD, space, w, i, t, h, space, start color blueD, 4, end color blueD, point, end color gray
equals, 5, plus, 12, space, space, space, space, space, space, space, space, start color gray, M, u, l, t, i, p, l, y, space, f, i, r, s, t, space, b, e, c, a, u, s, e, space, o, f, space, o, r, d, e, r, space, o, f, space, o, p, e, r, a, t, i, o, n, s, end color gray.
equals, 17
So, the expression 5, plus, 3, e equals 17 when e, equals, 4.
Notice how we had to be careful to think about order of operations when evaluating. A common wrong answer is start color redD, 32, end color redD, which comes from first adding 5 and 3 to get 8 then multiplying 8 by 4 to get start color redD, 32, end color redD.

Let's practice!

Evaluate the expression 9, minus, z when z, equals, 4.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

Evaluate the expression 9, minus, z when z, equals, 8.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

2, dot, 3, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

2, left parenthesis, 3, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

4, dot, 5, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

6, left parenthesis, 4, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

Evaluate 2, y when y, equals, 6.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

Evaluate the expression 7, plus, 4, b when b, equals, 3.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

Evaluate the expression start fraction, 8, divided by, d, end fraction, plus, 3 when d, equals, 4.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

Challenge problem 1

Evaluate e, dot, e, minus, 5, e when e, equals, 5.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

empty space, start color blueD, e, end color blueD, dot, start color blueD, e, end color blueD, minus, 5, start color blueD, e, end color blueD
equals, start color blueD, 5, end color blueD, dot, start color blueD, 5, end color blueD, minus, 5, dot, start color blueD, 5, end color blueD, space, space, space, space, space, space, space, space, start color gray, R, e, p, l, a, c, e, space, start color blueD, e, end color blueD, space, w, i, t, h, space, start color blueD, 5, end color blueD, point, end color gray
equals, 25, minus, 25, space, space, space, space, space, space, space, space, start color gray, M, u, l, t, i, p, l, y, space, f, i, r, s, t, space, b, e, c, a, u, s, e, space, o, f, space, o, r, d, e, r, space, o, f, space, o, p, e, r, a, t, i, o, n, s, end color gray.
equals, 0

Challenge problem 2

Complete the table to evaluate 2, x at different values of x.
x2, x
12
24
36
4
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i
5
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i
6
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i