Algebraic equations contain both numbers and variables. In this section, we will learn how to evaluate algebraic expressions when the values of the variables are given.
A mathematical expression that consists of variables, numbers and algebraic operations is called an algebraic expression.
For example: $30x+20y$ and $5x^3y-2xy^2-z+4$ are algebraic expressions.
Evaluate an algebraic expression means to find the value of the expression when the variables are substituted by certain numbers.
For example: when $x=2$ and $y=3$, the value of the expression $30x+20y$ is $30\cdot 2+20\cdot 3=60$.
Evaluating an Expression:
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Replace each variable by the given numerical value.
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Simplify the resulting expression. Be careful to follow the order of operations.
Examples: Evaluate for $a = 1$, $b = 2$, $c = 4$, and $d = -1$.
a) $5ab= 5(1)( 2) = 10$
b) $\dfrac{7b+c}{a-d}= \dfrac{7(2)+(4)}{1-(-1)}= \dfrac{14+4}{1+1}=9$
Examples: Evaluate for $a = -3$, $b = 5$, $c = -2$, and $d = 7$.
a) $4c-2b=4(-2)-2(5)=-8-10=-18$
b) $b^2+b = (5)^2+5 = 25+5=30$
c) $(c+a)(c^2-ac+a^2)$
$= ((-2)+(-3))((-2)^2-(-3)(-2) +(-3)^2) $
$= (-5)(4-6+9) $
$= (-5)(7) $
$= -35$
Example: Find the area of a triangle with height of $20\;\mathrm{in}$ and a base of $30\;\mathrm{in}$.
The area is $A=\dfrac{1}{2}\cdot b\cdot h = \dfrac{1}{2}\cdot 20 \;\mathrm{in }\cdot 30\;\mathrm{in} = 300 \;\mathrm{in}^2$.
Practice Problems:
1. Evaluate each algebraic expression for the given value(s).
(a) $x^2+8x$ for $x=6$
(b) $x^2-x+7$ for $x=3$
(c) $6+3(x-5)^3$ for $x=7$
(d) $x^2-3(x-y)$ for $x=7$ and $y=2$
2. Find the area of a circle with radius 7 cm, keep your answer with $\pi$. (Note that the area of a circle is $\pi r^2$.)
Answer Key: 1.(a) $84$ 1.(b) $13$ 1 .(c) $30$ 1 .(d) $34$ 2. $49\pi\;\mathrm{cm}^2$
For more detailed explanation, please read: Arithmetic|Algebra Chapter 4.
