
Linear
Equations and Word Problems 


Linear Equations
in One Variable

Solving linear
equations 





Linear
equation
in one variable

An
equation in x
is a statement that two algebraic expressions are equal. 
An equation that is
true for all values of its variables is called an identity
(identical equation). 
For example, (a
 b)(a +
b) = a^{2}  b^{2}. 
An
equation that is true for the certain value of the variable (the
root of the equation) is called a conditional equation. 

A
linear equation is any equation
that can be written in the form 
ax + b
= 0, where a
and b
are constants
(fixed real numbers) with a
not 0
and x
is a variable. 
A linear, means the
variable x appears only to the first power. 
The variable, as unknown quantity of which the values are to be found,
may be denoted other than x. 
To
solve an linear equation in x
means to find the value of x
for which the equation is true. Such value is a solution. 
For
example, x
= 2 is a solution of the
equation 3x 
6 = 0 because 3 · 2 
6 = 0
is a true statement. 

Solving linear
equations 
A
linear equation has exactly one solution. 
Simplify
both sides of the equation by removing
symbols of grouping (like brackets or fractions), combining like
terms, or simplifying fractions on one or both sides of the
equation. 
To
solve an equation, isolate its variable on the left side of the
equation by a sequence of equivalent simpler equations, each of
which have the same solution as the original (given) equation. 
To
get all terms with variable in them on one side of the equation
and all constants on the other side use the following methods: 
 add
or subtract the same quantity to or from each side of the
equation, 
 when the operations are addition or subtraction a term
may be shifted to the other side of an equation by changing its
sign, 
 multiply
or divide each side of the equation by the same nonzero
quantity, 
 divide (or multiply) both sides of the equation by the coefficient of the variable to make
its value unity. 
To
solve an equation involving fractional expressions, find the
least common denominator of all terms and multiply every term of
the equation by the common denominator. 
When
multiplying or dividing an equation by a variable quantity, it
is possible to introduce an extraneous solution. An extraneous
solution is one that does not satisfy the original equation. 
If there are variables in the denominators of the fractions
identify values of the variable which will give division by zero
as we have to avoid these values in the solution. 
After
solving an equation, verify the answer by plugging the result
into the original equation. 

Examples:
a)
23  [9x
 (17 + 5x)
 6]
= 10x  [11
 (2x
 7)
+ 21x] + 9 
23 
[9x 
17  5x 
6] = 10x 
[11 
2x + 7 + 21x]
+ 9 
23 
4x + 23
= 10x  19x
 18 + 9

5x = 
55  ¸5 
x = 
11 

b)
3(x
 1)
· (x
+ 5)
+ x · (x
 4)
= 4x(3
+ x)
+ 5 
3(x^{2}
+ 4x 
5) + x^{2
} 4x
= 12x + 4x^{2} + 5 
3x^{2} + 12x
 15
+ x^{2 } 4x
= 12x + 4x^{2} + 5 
 4x
= 20  ¸(
4) 
x = 
5 














Intermediate
algebra contents 



Copyright
© 2004  2020, Nabla Ltd. All rights reserved. 