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 be converting rational expressions (one quantity divided by another quantity) into linear expressions and square root equations into linear equations. The solution(s) to these converted equations might not be the same as the solution(s) to the original equation. After certain operations, you must check the solution(s) to the converted equation in the original equation.
To solve a rational equation, clear 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.
The following is a rational equation in
the form of one fraction equals another. We will use the fact that for b and d nonzero, 
if and only if ad = bc.
This method is called cross
multiplication.
Check: 
is a true
statement, so x ¼ 3 is the solution.
Anytime you multiply (or divide) both sides
of the equation by an expression with a variable in it, you must check your
solution(s) in the original equation. When you cross multiply, you are
implicitly multiplying both sides of the equations by the denominators of each
fraction, so you must check your solution in this case as well. The reason is
that sometimes a solution to the converted equation will cause a zero to be in
a denominator of the original equation. Such solutions are called extraneous
solutions. See what happens in the next example.
But x = –2
leads to a zero in a denominator of the original equation, so x = –2 is not a
solution to the original equation. The original equation has nosolution.
Have you ever wondered why expressions like 2/0
are not numbers? Let us see what complications arise when we try to see what ‘‘2/0’’
might mean. Say 2/0 = x.
Now cross
multiply.
2(1) = 0(x)
Multiplication
by zero always yields zero, so the
right hand side is zero.
2 = 0 No value for x
can make this equation true.
Or, if you
try to ‘‘clear the fraction’’ by multiplying both sides of the equation by a
common denominator, you will see that an absurd situation arises here, too.
So, 0 = 0x,
which is true for any x. Actually, the expression 0/0 is
not defined.
On some equations, you will want to raise
both sides of the equation to a power in order to solve for x. Be careful to
raise both sides of the equation to the same power, not simply the side with
the root. Raising both sides of an equation to an even power is another
operation which can introduce extraneous solutions. To see how this can happen,
let us look at the equation x = 4. If we square both sides of the equation, we
get the equation x2 = 16. This equation has two solutions: x = 4 and
x = –4.
Example
Remember
that 
(a if a is not negative. We will use this fact
t eliminate the square root sign. So, to ‘‘undo’’ a square root, first isolate the square root on one side of the equation
(in this example, it already is) then
square both sides.
Because we
squared both sides, we need to make sure x = 37 is a solution to the original
equation.
is a true
statement, so x = 37 is the solution.
Quadratic equations, to be studied in the
last chapter, have their variables squared—that is, the only powers on
variables are one and two. Some quadratic equations are equivalent to linear
equations.
Example
Because 36x2
is on each side of the equation, they cancel each other, and we are left with
–60x + 25 = 19x – 6,
an ordinary
linear equation.
Because we
neither multiplied (nor divided) both sides by an expression involving a variable nor raised both sides to
a power, it is not necessary to check your solution. For accuracy, however,
checking solutions is a good habit.
Practice
Solutions
Unless a
solution is extraneous, the check step is not printed.
“Sumber Informasi”
Thanks for reading Equations Leading to Linear Equations. Please share...!
