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College Mathematics Notes
Section 1.7
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Chapter 1: Pre-Algebra
Section 1.7: Evaluating and Simplifying Expressions
Big Idea for this section: Expressions with letter variables are used to represent the steps of a multi-step
calculation.
Big Skills for this section: You should be able to correctly evaluate or simplify a given expression with
variables.
Section 1.7.1 Evaluating a Variable Expression
When you are asked to evaluate a variable expression, replace the variables with given values, then follow the
order of operations to simplify the remaining numeric expression to a single value.
Use the rules of significant digits to correctly round calculations involving measurements.
Practice:
Evaluate the expression a 2  2b when a = 5 and b = 3.
Evaluate the expression y 2  2( x  7) when x = 4 and y = 9.
1
bh when b = 4.0 cm and h = 12.5 cm.
2
M
Evaluate the expression
when M = 127.58 g and V = 151.5 mL.
V
ft
ft
t  6.8 ft when t = 5.89 sec.
Evaluate the expression 32.0 2 t 2  16.5
sec
sec
Evaluate the expression
College Mathematics Notes
Section 1.7
Page 2 of 3
Section 1.7.2 Simplifying Expressions
Many times, we write down a variable expression to capture a formula for how to perform a multi-step
calculation, but that expression is not as efficient as it could be. Simplifying the expression before we have to
evaluate it allows for maximum efficiency when we finally do evaluate the expression, especially if we have to
evaluate it over and over again.
In this section, we will look at some basic simplification techniques involving addition, subtraction, and
multiplication in variable expressions.
Section 1.7.2 Addition and Subtraction of Variable Terms
Even if you have never worked with variables, you are probably already familiar with combining like terms
when dealing with units of measurement:
For example, the sum of 4 feet and 5 feet is 9 feet: 4ft + 5ft = 9ft
As another example, 3 feet 4 inches plus 7 feet 3 inches is: 3ft + 4in + 7ft + 3in = 10ft + 7in
Note that we can only add together like units of measurement.
Likewise, if we have variable expressions with different variables, we can only combine terms that have the
exact same variable.
4x + 5x = 9x
3a + 4b + 7a + 3b = 10a + 7b
The number in front of any variable is called the coefficient of the variable. The coefficient is the number
attached a variable term by multiplication. To add or subtract variable terms, we combine the coefficients and
keep the same variable (these are called like terms).
If two terms have different variable parts, then we cannot simplify the expression. For example, the expression
12b + 4w has no simpler form. How else could you write “12 bananas and 4 watermelons” without losing
information?
Also note that a constant term (one with no variable) is different from any variable term. If terms have different
variables, leave them alone. If they have the same variable, you may combine the coefficients.
Examples
6y + 10y simplifies to:
3t – 21t simplifies to:
62w – 70w simplifies to:
–20 + 4x – 3 =
8x – 3 + 7x + 9 =
9m + 3p – 4m + 12 – p =
College Mathematics Notes
Section 1.7
Page 3 of 3
Variable terms must also have the same exponent to qualify as like terms. 3x2 and 5x2 are like terms and may be
added: 3x2 + 5x2 = 8x2.
Note: The exponent is never changed by an addition or subtraction operation.
Example
Simplify 30t 2  7t – 8 –12t 2  3t  5 .
Section 1.7.3 Multiplication using the distributive property
The distributive property of real numbers shows how multiplication and addition/subtraction interact. For
example, we can see that 4(2 + 3) = 4(5) = 20 if we simply follow the order of operations. On the other hand,
we can multiply first provided we multiply through to each term inside the parentheses: 4(2 + 3) = 4(2) + 4(3) =
8 + 12, which is 20 again. This property always works, no matter what the numbers are.
In general, we can write the distributive property as a(b + c) = ab + bc for all real numbers a, b, and c. In other
words, when we have a multiplier in front of a parentheses group, we can choose to simplify and remove the
parentheses provided that we multiply through to each term inside the parentheses. This is especially useful
when variables are involved, because we may have no other way to simplify an expression with parentheses.
Practice:
2(x + 3) =
5(p – 9) =
3(4c – 2) =
7(2x + 1) =
2( f  6) =
4(9v  2) =
–6  2 x – 5 y –10  =
2(4 x  9)  3(2 x  1) =
2(m  7)  5(2m  3) =