Algebra Demystified 2nd Ed

210 algebra De mystif ieD

Equations Leading to Linear Equations

Some equations are almost linear equations; after one or more steps, these

equations become linear equations. In this section, we will convert rational

equations (involving fractions containing variables) into linear equations

and equations containing a square root into linear equations. After certain

operations involving variables, though, we must check the solution(s) to the

converted equation in the original equation.

To solve a rational equation, we begin by clearing the fraction. In this book,

two approaches will be used. First, if the equation is in the form of “fraction =

fraction,” cross-multiply to eliminate the fraction. Second, if there is more

than one fraction on one side of the equal sign, the LCD will be determined

and each side of the equation will be multiplied by the LCD. These are not

the only methods for solving rational equations, but they are usually the

quickest.

The following is a rational equation in the form of one fraction equals

another. We use the fact that for b and d nonzero, ab = dc if and only if ad = bc.

This method is called cross-multiplication.

EXAMPLE

Solve the equation.

1

1

1

x− 2

=

We cross-multiply by multiplying the left side of the equation by 2, the

denominator on the right side, and multiplyingthe right side of the

equation by x – 1, the denominator on the left side.

1

1

1

2

21 11

x

x

−

=

()=−()(Thisisthecross-multiplicattionstep.)

21

11

3

=−

++

=

x

x

We now check our solution.

1

31

1

2 

– 

=  is a true statement, so x = 3 is the solution.

EXAMPLE

Solve the equation.

EXAMPLE

Solve the equation.