48 CHAPTER 2 Linear Equations, Inequalities, and Applications
OBJECTIVES OBJECTIVE 1 Distinguish between expressions and equations. In our work
in Chapter 1,we reviewed algebraic expressions.
and Examples of algebraic expressions
Equations and inequalities compare algebraic expressions, just as a balance scale
compares the weights of two quantities. Recall from Section 1.1that an equationis a
statement that two algebraic expressions are equal. An equation always contains an
equals symbol, while an expression does not.
Distinguishing between Expressions and Equations
Decide whether each of the following is an expressionor an equation.
(a) (b)
In part (a) we have an equation, because there is an equals symbol. In part (b),
there is no equals symbol, so it is an expression. See the diagram below.
Left Right
side side
Equation Expression
(to solve) (to simplify or evaluate)
NOW TRY
OBJECTIVE 2 Identify linear equations, and decide whether a number is
a solution of a linear equation.A linear equation in one variableinvolves only
real numbers and one variable raised to the first power.
x+ 1 =-2, x- 3 =5, and 2 k+ 5 = 10 Examples of linear equations
3 x- 7 = 2 3 x- 7
3 x- 7 = 2 3 x- 7
EXAMPLE 1
x^3 y^8
z
8 x+9, y- 4,
Linear Equations in One Variable
2.1
1 Distinguish between
expressions and
equations.
2 Identify linear
equations, and
decide whether a
number is a solution
of a linear
equation.
3 Solve linear
equations by using
the addition and
multiplication
properties of
equality.
4 Solve linear
equations by using
the distributive
property.
5 Solve linear
equations with
fractions or
decimals.
6 Identify conditional
equations,
contradictions, and
identities.
⎧⎪⎨⎪⎩ {
Linear Equation in One Variable
A linear equation in one variablecan be written in the form
where A, B, and Care real numbers, with AZ 0.
AxBC,
A linear equation is a first-degree equation,since the greatest power on the vari-
able is 1. Some equations that are not linear (that is, nonlinear) follow.
and Examples of nonlinear equations
If the variable in an equation can be replaced by a real number that makes the
statement true, then that number is a solutionof the equation. For example, 8 is a so-
lution of the equation since replacing xwith 8 gives a true statement. An
equation is solvedby finding its solution set,the set of all solutions. The solution set
of the equation x- 3 = 5 is 586.
x- 3 =5,
2 x= 6
8
x
x^2 + 3 y=5, =-22,
NOW TRY
EXERCISE 1
Decide whether each of the
following is an expressionor
an equation.
(a)
(b) 2 x+ 17 = 3 x
2 x+ 17 - 3 x
NOW TRY ANSWERS
- (a)expression (b)equation