To evaluate an algebraic expression, we must first replace or substitute the variables in the expression with numbers, and then we simplify the expression by applying the Order of Operations. If, after we have evaluated, there are still variables, then we should be flagged that we haven't completely solved the problem. Since each variable in the expression is replaced with a numeric value, our final solution must also take the form of a numeric value.

Example 1: Evaluate \(2x - 3\) when \(x = 6\)

First, substitute \(x\) with \(6\) in the expression. Then, simplify using the Order of Operations.

$$\eqalign{

2x - 3 &= 2\left( 6 \right) - 3 \cr

& = 12 - 3 \cr

& = 9 \cr} $$

Example 2: Evaluate \(a + {b^2}\) when \(a=-5\) and \(b=3\)

Replace the variables with the assigned numerical values then simplify. Note, the term \(b^2\) implies that we are going to multiply the value of \(b\) by itself.

$$\eqalign{

a + {b^2} &= - 5 + {\left( 3 \right)^2} \cr

& = - 5 + 9 \cr

& = 4 \cr} $$

Example 3: Evaluate \(\large{{{2x - 4y} \over {x + y}}}\) if \(x=3\) and \(y=-1\)

When you evaluate this expression, make sure to correctly apply the rules for integer addition and integer multiplication.

$$\eqalign{

{{2x - 4y} \over {x + y}} &= {{2\left( 3 \right) - 4\left( { - 1} \right)} \over {3 + \left( { - 1} \right)}} \cr

& = {{6 + 4} \over 2} \cr

& = {{10} \over 2} \cr

& = 5 \cr} $$