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 |