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Linear
equation
in one variable
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Linear
equation |
Solving linear equations |
Linear
equation
in one variable examples
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Linear
equation
in one variable
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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) = a2 - b2. |
An
equation that is true for the certain value of the variable (the
root of the equation) is called a conditional equation. |
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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. |
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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. |
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Linear
equation
in one variable examples
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Examples:
a)
-1
-
(2
-
x)
= 2
-
(3x
+
1) |
-1
-
2
+
x
= 2
-
3x
-
1 |
x
+
3x
= 2 +
2 |
4x
= 4
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¸
4 |
x
= 1 |
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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
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5x = -
55 | ¸5 |
x = -
11 |
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b) 3(x
- 1)
· (x
+ 5)
+ x · (x
- 4)
= 4x(3
+ x)
+ 5 |
3(x2
+ 4x -
5) + x2
- 4x
= 12x + 4x2 + 5 |
3x2 + 12x
- 15
+ x2 - 4x
= 12x + 4x2 + 5 |
- 4x
= 20 | ¸(-
4) |
x = -
5 |
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Functions
contents A
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