©
:
Pre

Calculus

Chapter 2A
Chapter 2A

Solving Equations
Introduction and Review of Linear Equations
An equation is a statement which relates two or more numbers or algebraic expressions. For example,
the equation
2
4
6
is a statement relating the equality of the expression 2
4 with the number 6. This statement is always
true. The equation
x
3
12
is a statement relating the equality of the expression
x
3 with the number 12. This statement may or
may not be true depending on what numerical value the variable
x
is assigned.
A solution is a value of the variable which makes the equation true. It can be seen by inspection that
x
9 is a solution of the above equation since
9
3
12
To verify that a number is a solution to an equation, replace the variable with that value. If the
resulting statement is true, the value is a solution, otherwise, it is not a solution.
We will explore methods for solving many different kinds of equations in this section. The simplest
kind is the linear equation, which is an equation that can be written in the form
ax
b
0
This kind of equation is solved by isolating the variable, namely, undoing what has already been done
to it.
Example
:
Solve the equation 6
3
x
x
−
4 for
x
.
Solution:
Begin by moving all the variable expressions to one side of the equation and all the
numbers to the other side:
6
3
x
x
−
4
subtract 6 from both sides
3
x
6
−
6
x
−
4
−
6
Note the commutative property
3
x
x
−
10
now subtract
x
from both sides
3
x
−
x
x
−
x
−
10
x
−
x
cancels to 0
2
x
−
10
divide both sides by 2
x
−
5
Final solution
We can check our solution by replacing
x
with
−
5 in the original equation:
6
3
−
5
−
5
−
4 ?
6
−
15
−
9 ?
−
9
−
9
Therefore, our solution is correct.
©
:
Pre

Calculus
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